This is a multi-part message in MIME format. ---------------0306030901183 Content-Type: text/plain; name="03-254.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-254.comments" With the Appendix "Families of periodic orbits in reversible PDEs", written by Dario Bambusi ---------------0306030901183 Content-Type: text/plain; name="03-254.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-254.keywords" Hamiltonian PDE, symplectic structure, Lax-integrable equation, KAM-theory, Nekhoroshev theorem, Gibbs measure, Gromov theorem, symplectic capacity, squeezing, periodic solutions ---------------0306030901183 Content-Type: application/postscript; name="obzor2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="obzor2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: handb.dvi %%Pages: 24 0 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %%BeginProcSet: PStoPS 1 15 userdict begin [/showpage/erasepage/copypage]{dup where{pop dup load type/operatortype eq{1 array cvx dup 0 3 index cvx put bind def}{pop}ifelse}{pop}ifelse}forall [/letter/legal/executivepage/a4/a4small/b5/com10envelope /monarchenvelope/c5envelope/dlenvelope/lettersmall/note /folio/quarto/a5]{dup where{dup wcheck{exch{}put} {pop{}def}ifelse}{pop}ifelse}forall /setpagedevice {pop}bind 1 index where{dup wcheck{3 1 roll put} {pop def}ifelse}{def}ifelse /PStoPSmatrix matrix currentmatrix def /PStoPSxform matrix def/PStoPSclip{clippath}def /defaultmatrix{PStoPSmatrix exch PStoPSxform exch concatmatrix}bind def /initmatrix{matrix defaultmatrix setmatrix}bind def /initclip[{matrix currentmatrix PStoPSmatrix setmatrix [{currentpoint}stopped{$error/newerror false put{newpath}} {/newpath cvx 3 1 roll/moveto cvx 4 array astore cvx}ifelse] {[/newpath cvx{/moveto cvx}{/lineto cvx} {/curveto cvx}{/closepath cvx}pathforall]cvx exch pop} stopped{$error/errorname get/invalidaccess eq{cleartomark $error/newerror false put cvx exec}{stop}ifelse}if}bind aload pop /initclip dup load dup type dup/operatortype eq{pop exch pop} {dup/arraytype eq exch/packedarraytype eq or {dup xcheck{exch pop aload pop}{pop cvx}ifelse} {pop cvx}ifelse}ifelse {newpath PStoPSclip clip newpath exec setmatrix} bind aload pop]cvx def /initgraphics{initmatrix newpath initclip 1 setlinewidth 0 setlinecap 0 setlinejoin []0 setdash 0 setgray 10 setmiterlimit}bind def end %%EndProcSet %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o paper.ps handb %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.03:0954 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (handb.dvi) @start %DVIPSBitmapFont: Fa cmtt10 10 21 /Fa 21 120 df<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B708A2C>46 D<1507ED0F80151FA2153F16005D157E15FE5D14015D14035DA214075D140F5D141F5D14 3F92C7FC5C147E14FE5CA213015C13035C13075C130F5C131F5CA2133F91C8FC5B137E13 FE5B12015B12035B12075BA2120F5B121F5B123F90C9FC5A127E12FE5AA25A127821417B B92C>I<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F00C7FCAE121FEA3F80EA7FC0 EAFFE0A5EA7FC0EA3F80EA1F000B2470A32C>58 D64 D<3801FFF0000713FE001F6D7E15E048809038C01FF81407EC01FC381F80000006C77EC8 127EA3ECFFFE131F90B5FC1203120F48EB807E383FF800EA7FC090C7FC12FE5AA47E007F 14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001EBFE1F3A003FF007FC27247CA32C> 97 DI<903803FFE0011F13F8017F13FE48B5FC48804848C6FCEA0FF0485A49137E 4848131890C9FC5A127EA25AA8127EA2127F6C140F6DEB1F806C7E6D133F6C6CEB7F0039 07FE03FF6CB55A6C5C6C6C5B011F13E0010390C7FC21247AA32C>III104 D<1307EB1FC0A2497EA36D5AA20107C7FC90C8FCA7387FFFC080B5FC7EA2EA0007B3A800 7FB512FCB612FEA36C14FC1F3479B32C>I107 D<3A7F83F007E09039CFFC1FF83AFFDF FE3FFCD87FFF13FF91B57E3A07FE1FFC3E01FCEBF83F496C487E01F013E001E013C0A301 C01380B33B7FFC3FF87FF0027F13FFD8FFFE6D13F8D87FFC4913F0023F137F2D2481A32C >109 D<397FF01FE039FFF87FFC9038F9FFFE01FB7F6CB6FC00019038F03F80ECC01F02 807FEC000F5B5BA25BB3267FFFE0B5FCB500F11480A36C01E0140029247FA32C>I I<397FF01FE039FFF8FFF801FB13FE90B6FC6C158000019038F07FC09138801FE0913800 07F049EB03F85BED01FC491300A216FE167EA816FE6D14FCA2ED01F86D13036DEB07F015 0F9138801FE09138E07FC091B51280160001FB5B01F813F8EC3FC091C8FCAD387FFFE0B5 7EA36C5B27367FA32C>I114 D<90387FF8700003B512F8120F5A5A387FC00F387E00034813015AA36CEB 00F0007F140013F0383FFFC06C13FE6CEBFF80000314E0C66C13F8010113FCEB0007EC00 FE0078147F00FC143F151F7EA26C143F6D133E6D13FE9038F007FC90B5FC15F815E000F8 148039701FFC0020247AA32C>I<131E133FA9007FB6FCB71280A36C1500D8003FC8FCB1 ED03C0ED07E0A5EC800F011FEB1FC0ECE07F6DB51280160001035B6D13F89038003FE023 2E7EAD2C>I<3A7FF003FF80486C487FA3007F7F0001EB000FB3A3151FA2153F6D137F39 00FE03FF90B7FC6D15807F6D13CF902603FE07130029247FA32C>I119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy6 6 1 /Fb 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc eufm10 10 1 /Fc 1 66 df<1840DAFF8014C0010F01F0EB0380013F01FCEB070090B56C130E2701E03F FF133E2603800FEB807E260F0003EB81FC000E6D13C1487F003C147F48EC3FE1A200F814 1FA27E6C140F7E7F6C7EA26C6C14C17F121F6C7E000715810003141FA26C481401153F49 133E4848137E49137C48C712F8000EEB01F0003CEB03E00010EB0780C7EA0F00141E5C5C 5C495A494880EB0F80EB1FE0D93FF8EB07FFD9FFFC131F486D013D1380486D13F848DA81 E013C3D81F039039C7C07FCFD83C00D9FF8013FE007090267FFE0013F800406D48EB3FF0 C76C4814E0DA0FE0EB1F806E4814004B130E0202C71208383C7EB93C>65 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmex7 7 1 /Fd 1 83 df<153EEDFF80913801C1C0913803806091380781E091380F03F0A2141EA2ED 01E0023EC7FCA4147E147CA614FCA7495AA9495AA75CA613075CA400785B12FCA249C8FC A2EA781EEA601CEA3838EA1FF0EA07C024417C7F20>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe msam10 10 1 /Fe 1 99 df<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FECAFCEA01F8 485A485A485A5B48CBFCA2003E0103B61280020F15C048133F4A1580007801FEC9FC38F8 01F85C48485AA25CA480A26C6C7E80387800FE007C017FB612806E15C06C130F02031580 6C90CAFCA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F15 80323279AD41>98 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmti8 8 5 /Ff 5 112 df97 D99 D<153EEC07FEA2EC007EA2157CA215FCA215F8 A21401A215F0A21403EB07C390381FF3E0EB7C3BEBF81FEA01F03903E00FC0EA07C0120F EA1F801580EA3F00141F5A007E1400A25C12FE48133EA2EC7E18153848137CA214FCD878 011378397C03F870A2393C0F78E0381E1E3D390FF81FC03903E00F001F2F79AD24>I<13 1FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA21201147E9038F3FF809038F787 C03903FE03E013FC13F8A2EA07F013E0A213C0000F130715C01380A2001F130F15801300 141F481406150E003E133F143E007E141EEC7E1C007C137CEC3C3812FC157048EB1FE000 70EB07801F2F7BAD24>104 D111 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi6 6 4 /Fg 4 111 df<140C141C143C1438A21478147014F014E0130114C0A213031480130714 00A25B130E131E131CA2133C13381378137013F05BA212015B12035BA2120790C7FC5A12 0EA2121E121C123C123812781270A212F05AA216317CA420>61 D<903AFFFE07FFF0A290 3A07C0003E00A249485BA449C75AA4013E495AA3013FB5FC495C90387C0003A349495AA4 4848495AA4484849C7FCA300075C3AFFFE07FFF0A22C227CA132>72 D<903801FFFCA290380007C0A2EC0F80A4EC1F00A4143EA45CA45CA4495A1218123C127E 48485AA248485A38600F80D8783EC7FCEA3FFCEA0FE01E237CA122>74 D<000F13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E03903C00780A3EC0F 00EA0780A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE0000CEB07801F17 7D9526>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh msbm8 8 3 /Fh 3 85 df78 D82 D84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmbx7 7 1 /Fi 1 87 df86 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy8 8 2 /Fj 2 51 df<170EA3170F8384170384170184717E1878187C84180FF007C0BA12F819FC 19F8CBEA07C0F00F00183E601878604D5A60170360170795C7FC5F170EA33E237CA147> 33 D<91B512C01307131FD97F80C7FC01FCC8FCEA01F0EA03C0485A48C9FC120E121E5A 123812781270A212F05AA3B712C0A300E0C9FCA37E1270A212781238123C7E120E120F6C 7E6C7EEA01F0EA00FCEB7F80011FB512C013071300222B7AA52F>50 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmmi8 8 16 /Fk 16 121 df14 D23 D<1560A315E0A25DA21401A25DA21403A292C7FCA25CEC3FE0903803FFF890380FC63E90 393E0E0F80017CEB07C03A01F00C03E0D803E0EB01F03807C01CD80F801300001F011813 F81300003E1338A2481330A2EC700100FC15F0481360150302E013E01507007801C013C0 007CEC0F800101EB1F00003C143E003E495A001F5C390F8383E03903E39F802600FFFEC7 FCEB1FF00107C8FCA21306A2130EA2130CA2131CA21318A3253C7DAD2A>30 D<160E486C143F120348C813801206000E151F000C150F001C160000188112380030130C 141E007015061260143E023C130E00E0150C5A0238131C6C01785B14705E02F013F06C48 6C485A010313033A7C0FFC0FC03A7FFFBFFF80023F90C7FC393FFC1FFE391FF80FF83907 E007E0291F7F9D2C>33 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206 120E5A5A5A126009157A8714>59 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80 EB0FE0EB03F8EB00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE 1FC01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14 FCEB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F8000FECAFC12F812E02A2B7AA537 >62 D<90273FFFFC0FB5FCA2D900FEC7EA3F80A24A1500A201015D177E5CA2010315FE5F 5CA2010714015F5CA2010F14035F5C91B6FC5B9139C00007E05CA2013F140F5F91C7FCA2 49141F5F137EA201FE143F94C7FC5BA200015D167E5BA2000315FEB539E03FFFF8A2382D 7CAC3A>72 D<000FB8FCA23B1FC003F8003F0100151F001C4A130E123C00380107140612 3000704A130EA20060010F140C12E0485CA2141FC715005DA2143FA292C8FCA25CA2147E A214FEA25CA21301A25CA21303A25CA21307A25C130F131F001FB512F0A2302D7FAC29> 84 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007 151C16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E12 7EA2150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC38 03E03C3801FFF038003FC02F2E7BAC30>I<90260FFFFCEB7FFFA29026007FC0EB0FF06E 48148018006E6C131E1718020F5C6F5B02075C6F485A020349C7FCEDF8065E6E6C5A5E6E 6C5A5EED7F8093C8FC6F7EA26F7E153F156FEDCFE0EC018791380307F0EC0703020E7F14 1C4A6C7E14704A6C7E495A4948137F49C7FC010E6E7E5B496E7E5BD801F081D807F8143F D8FFFE0103B5FCA2382D7EAC3A>88 D<151FEC03FFA2EC003FA2153EA2157EA2157CA215 FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F03903E007E0EA07C0120FEA1F 8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F021506143E5AEC7E0E007CEB FE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903F003C0202F7DAD24>100 D<3907C007E0391FE03FF83918F8783E393879E01E39307B801F38707F00126013FEEAE0 FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF8081618EBC00115F0000F 1538913803E0300180147016E0001F010113C015E390C7EAFF00000E143E251F7E9D2B> 110 D115 D<130E131FA25BA2133EA2137EA2137CA213FCA2B512F8A23801F800A25BA21203A25BA2 1207A25BA2120FA25BA2001F1310143013001470146014E0381E01C0EB0380381F0700EA 0F0EEA07FCEA01F0152B7EA919>II<013F137C9038FFC1FF3A01C1E383803A 0380F703C0390700F60F000E13FE4813FC12180038EC0700003049C7FCA2EA200100005B A313035CA301075B5D14C000385CD87C0F130600FC140E011F130C011B131C39F03BE038 D8707113F0393FE0FFC0260F803FC7FC221F7E9D28>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr8 8 62 /Fl 62 124 df<14FF010713E090381F80F090383E003849137C4913FC485A1203491378 153092C7FCA7157CB612FCA23803E000157CB3A5486C13FE3A7FFF0FFFE0A2232F7FAE27 >12 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009 157AAD14>39 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2 121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00 F013701338131C130E1307130310437AB11B>I<12C07E12707E7E7E120FEA0780120313 C0EA01E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A3 13F0A2EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D<123C127EB4FCA21380A2127F123D1201 A312031300A25A1206120E5A5A5A126009157A8714>II<123C12 7E12FFA4127E123C08087A8714>I48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E143EA2147E14FEA2EB01BE1303143E13 06130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812305A 12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90 B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380 000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C0007013 0F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>I<1230123C003F B512F8A215F05A15E039700001C000601480140348EB0700140E140CC7121C5C14301470 5C495AA2495AA249C7FCA25B130E131EA2133EA3133C137CA413FCA913781D2E7CAC23> 55 DII<123C127E12FFA4127E123C1200AD12 3C127E12FFA4127E123C081D7A9C14>I61 D<4A7E4A7EA34A7EA24A7EA3EC1BF81419A2EC30FCA2EC70FEEC607EA24A7EA349486C7E A2010380EC000FA201066D7EA3496D7EA2011FB57EA29038180001496D7EA349147EA201 E0147F4980A20001ED1F801203000716C0D80FF0EC3FE0D8FFFC0103B5FCA2302F7EAE35 >65 DI68 DIIIII75 DIII 80 D<90383F80303901FFF0703807C07C390F000EF0001E130748130348130114001270 00F01470A315307EA26C1400127E127FEA3FE013FE381FFFE06C13FC6C13FF00011480D8 003F13E013039038003FF0EC07F81401140015FC157C12C0153CA37EA215787E6C14706C 14F06CEB01E039F78003C039E3F00F0038E07FFE38C00FF01E2F7CAD27>83 D<007FB712F8A29039000FC003007C150000701638A200601618A200E0161CA248160CA5 C71500B3A94A7E011FB512E0A22E2D7EAC33>I91 D93 D<12035A120E5A5A123012701260A212E05AA312DEB4FC1380A2127FA2EA3F00121E0915 7BAD14>96 D<13FF000713C0380F01F0381C00F8003F137C80A2143F001E7FC7FCA4EB07 FF137F3801FE1FEA07F0EA1FC0EA3F80EA7F00127E00FE14065AA3143F7E007E137F007F EBEF8C391F83C7FC390FFF03F83901FC01E01F207D9E23>III<15F8141FA214011400ACEB0FE0EB7FF838 01F81E3803E0073807C003380F8001EA1F00481300123E127EA25AA9127C127EA2003E13 017EEB8003000F13073903E00EFC3A01F03CFFC038007FF090391FC0F800222F7EAD27> III<013F13F890 38FFC3FE3903E1FF1E3807807C000F140C391F003E00A2003E7FA76C133EA26C6C5A0007 1378380FE1F0380CFFC0D81C3FC7FC90C8FCA3121E121F380FFFF814FF6C14C04814F039 1E0007F848130048147C12F848143CA46C147C007C14F86CEB01F06CEB03E03907E01F80 3901FFFE0038003FF01F2D7E9D23>II< EA0780EA0FC0EA1FE0A4EA0FC0EA0780C7FCA8EA07C012FFA2120F1207B3A5EA0FE0EAFF FCA20E2E7EAD14>I107 DI<2607C07FEB07F03BFFC3FFC03FFC903AC783F0783F3C0FCE 01F8E01F803B07DC00F9C00F01F8D9FF8013C04990387F000749137EA249137CB2486C01 FEEB0FE03CFFFE0FFFE0FFFEA2371E7E9D3C>I<3807C0FE39FFC3FF809038C703E0390F DE01F0EA07F8496C7EA25BA25BB2486C487E3AFFFE1FFFC0A2221E7E9D27>II<3807C0FE39FFC7FF809038CF03E0390FDC01F03907F800FC49 137E49133E49133FED1F80A3ED0FC0A8151F1680A2ED3F00A26D137E6D137C5D9038FC01 F09038CE07E09038C7FF80D9C1FCC7FC01C0C8FCA9487EEAFFFEA2222B7E9D27>I<9038 0FE01890387FF8383801F81C3903E00E783807C007390F8003F8001F1301EA3F00A2007E 1300A212FE5AA8127EA36C13017EEB8003380FC0073803E00E3801F03C38007FF0EB1FC0 90C7FCA94A7E91381FFFC0A2222B7E9D25>I<380781F838FF87FEEB8E3FEA0F9CEA07B8 13B0EBF01EEBE000A45BB0487EB5FCA2181E7E9D1C>I<3801FE183807FFB8381E01F8EA 3C00481378481338A21418A27E7EB41300EA7FF06CB4FC6C13C06C13F0000113F838001F FC130138C0007E143EA26C131EA27EA26C133CA26C137838FF01F038E3FFC000C0130017 207E9E1C>I<1360A413E0A312011203A21207121FB512F0A23803E000AF1418A7143838 01F03014703800F860EB3FE0EB0F80152A7FA81B>II<3AFFFC01FFC0A23A0FE0007E000007147C15380003143015706C6C1360A26C6C 5BA390387C0180A26D48C7FCA2EB3F07EB1F06A2EB0F8CA214DCEB07D8A2EB03F0A36D5A A26D5A221E7F9C25>I<3BFFFC3FFE07FFA23B0FE003F001F801C09038E000F000070101 14E0812603E00314C0A2913807F8012701F006781380A29039F80E7C030000D90C3C1300 A290397C181E06A2151F6D486C5AA2168C90391F600798A216D890390FC003F0A36D486C 5AA36DC75A301E7F9C33>I<3AFFFC07FF80A23A0FF003FC000003EB01F0000114C06D48 5A000091C7FCEB7C06EB3E0E6D5A14B8EB0FB0EB07E013036D7E497E1307EB067C497EEB 1C1F01387FEB700F496C7E6E7ED803C07F00076D7E391FE003FC3AFFF007FFC0A2221D7F 9C25>I<3AFFFC01FFC0A23A0FE0007E000007147C1538000314306D137000011460A26C 6C5BA2EBFC01017C5BEB7E03013E90C7FCA2EB1F06A2148EEB0F8CA2EB07D8A2EB03F0A3 6D5AA26D5AA2495AA2130391C8FC1278EAFC06A25B131CEA7838EA7070EA3FE0EA0F8022 2B7F9C25>I123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr6 6 10 /Fm 10 58 df<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F813 1FAD0078131EA2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227D A01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>II<13FF000313C0380F03E0381C00F014F8003E13FC147C A2001E13FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A2380003E0EB00F01478147C 143E143F1230127812FCA2143E48137E0060137C003813F8381E03F0380FFFC000011300 18227DA01E>I<14E01301A213031307A2130D131D13391331136113E113C1EA01811203 EA07011206120C121C12181230127012E0B6FCA2380001E0A6EB03F0EB3FFFA218227DA1 1E>I<00101330381E01F0381FFFE014C01480EBFE00EA1BF00018C7FCA513FE381BFF80 381F03C0381C01E0381800F014F8C71278A2147CA21230127812F8A214784813F8006013 F0387001E01238381E07803807FF00EA01F816227CA01E>II<1230123C003FB5FCA24813FE14FC3860001C1438 14704813E014C0EA0001EB0380EB07001306130E5BA25BA21378A35BA41201A76C5A1823 7CA11E>I<137F3803FFC0380781E0380E00704813380018131C1238A3123C003F133838 1FC078EBE0F0380FF9E03807FF80120114C0000713F0380F0FF8381C03FC383801FE3870 007E141F48130F1407A314060070130E0078130C6C1338001F13F03807FFC0C613001822 7DA01E>I<13FE3803FFC0380781E0380E0070481378003C133848133CA200F8131EA314 1FA40078133FA26C137F121C380F01DF3807FF9F3803FE1EC7FCA2143E143C001C133800 3E13781470003C13E0381801C0381C0780380FFE00EA03F818227DA01E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmmi5 5 13 /Fn 13 116 df<137F3803FFC04813E0EA0F010018C7FC5AA2EA39F8EA1FFC5BEA33F000 60C7FCA25AA26C13C038700380383FFF006C5AEA07F813147D921C>34 D65 D<3A03FFF03FFFA23A003E0003E0A449EB07C0A449EB0F80A290B6FCA23A01F0001F00A4 4848133EA448485BA43AFFFC0FFFC0A2281C7C9B2E>72 D<90380FFFC0A29038007C00A4 5CA4495AA4495AA4495AA21218127C38FC0F80A249C7FCEAF81EEA707CEA3FF8EA0FC01A 1D7B9B20>74 D<0003B512E015FC39003E003FED0F80ED07C0A25BA3ED0F8049EB1F0015 3E15F890B512E05A9038F003F0EC00F8A2485AA44848485AA2163016603AFFFC00F8E0ED 7FC0C8EA1F00241D7C9B2B>82 D97 DI100 D106 D108 D<3A0F01F807E03A3F87FE1FF8 3A33CE1F387C3A63D80F603CD8C3F013C001E01380D803C01300A22607801E5BA3EEF040 48484814C0ED01E0EEE18016E3001E90397800FF00000C0130137C2A127D9133>I<380F 03F0383F87FC3833DC1EEA63F8EAC3F013E0EA03C0A248485AA3EC7820D80F00136014F0 15C014F1001EEB7F80000CEB3E001B127D9125>I<137E3801FF80EA0381380703C0380E 0780EB0300EA0F80EA07F86CB4FC6C1380EA000FEA3003127812F8EB0700EAF00EEA7FFC EA1FF012127C911C>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo msbm7 7 4 /Fo 4 124 df82 D84 D<003FB612C0A23A33E0038180D83F001303003C903807070048EB06060070EB0E0EEC1C 0C0060EB181CEC3838C7EA7030EC6070ECE0E049485A148190380383800203C7FC495AEB 0E06EB0C0EEB1C1CEB3818EB3038EB7070EBE060EBC0E03801C1C0D981801360EA0383D8 0703C7FCD8060714E0EA0E0ED81C0C1301D8181C1303D83838EB07C0D87030130ED86070 131CD8E0E013F8B7FCA223287EA737>90 D<013FEC03E0D9FF80EB07F048ED1FE048ED7F C048913801FF800101903803FE000006EC0FF8D80003EB3FE04AB45A5CD907075B902606 1FF3C7FC90380E3FE790380CFF8690381FFE06ECFC0E90383FF00CEB7FC03A01FF001C07 D807FEEB180ED80FF8EB1FFED83FE05CD87F805C48C75B007CEC0FC02C197D982D>123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmti10 10 79 /Fp 79 128 df<04FFEB03F003039038E00FFC923A0FC0F01F1E923A3F00783E0F923A7E 01F87C3FDB7C03EBFC7F03FC14F8DA01F813F905F1137EDC01E1133C913B03F00003F000 A314074B130760A3140F4B130F60A3010FB812C0A3903C001F80001F8000A3023F143F92 C790C7FCA44A5C027E147EA402FE14FE4A5CA413014A13015FA313034A13035FA313074A 495AA44948495AA44948495AA3001CD9038090C8FC007E90380FC03F013E143E00FE011F 5B133C017C5C3AF8780F01E0D878F0EB07C0273FE003FFC9FC390F8000FC404C82BA33> 11 DI< EE7FE0923903FFFC7E92380FC03E92381F000F033EEB3FFE4B137F03FC14FC5D1401173D 4A48EB01F8A21703A24A4814F0A21707A2020F15E05D170FA218C0010FB7FCA3903B001F 80001F80A2173F143F92C71300A25FA24A147E147E17FEA25F14FE4A1301A25FA2010114 035CEFF070A21607010316F04AECE0E0A3EFE1C013074A14C3933803E380EE01E7933800 FF004948143C94C7FCA3495AA3001C90CAFC127E133E12FE133C137CEAF878EA78F0EA3F E0EA0F80374C82BA31>II<127812FCA27EA2 7E7E7EEA1F80120F13C01207EA03E01201120013C00B1068B92A>18 D<130FEB1F80133F137FEBFF00485A5BEA03F0485A485A485A003EC7FC5A5A12E05A1110 64B92A>I38 DI<150C151C153815F0EC01E0EC03C0EC 0780EC0F00141E5C147C5C5C495A1303495A5C130F49C7FCA2133EA25BA25BA2485AA212 035B12075BA2120F5BA2121FA290C8FCA25AA2123EA2127EA2127CA412FC5AAD1278A57E A3121C121EA2120E7EA26C7E6C7EA212001E5274BD22>I<140C140E80EC0380A2EC01C0 15E0A2140015F0A21578A4157C153CAB157CA715FCA215F8A21401A215F0A21403A215E0 A21407A215C0140F1580A2141F1500A2143EA25CA25CA2495AA2495A5C1307495A91C7FC 5B133E133C5B5B485A12035B48C8FC120E5A12785A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214F0150579941E>I<120EEA3F80127F12 FFA31300127E123C0909778819>I48 D<15181538157815F0140114031407EC0FE0141F147FEB03 FF90383FEFC0148FEB1C1F13001580A2143FA21500A25CA2147EA214FEA25CA21301A25C A21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7FC497EB61280A31D3877 B72A>III<16E0ED01F01503A3150716E0A3150F16C0 A2151F1680A2ED3F00A3157EA2157C15FC5D14015D14035D14075D140F5D141F92C7FC14 3EA25CECF81C153E903801F07EEB03E014C090380780FE130F49485A133EEB7C01137801 F05BEA01E03803C003EA0FFE391FFFC3F04813FB267C01FF13403AF0003FFFE000601307 C71400EC0FE05DA3141F5DA3143F92C7FCA4143E141C24487DB72A>I<010314186E13F8 903907F007F091B512E016C01600495B15F8010E13E0020CC7FC011EC8FC131CA3133C13 38A313781370A2147F9038F3FFC09038EF83E09038FC01F0496C7E485A497F49137CC8FC 157EA315FEA41401000C5C123F5A1403485C5A4A5A12F800E05C140F4A5A5D6C49C7FC00 70137E00785B387C01F8383E07F0381FFFC06C90C8FCEA01F8253A77B72A>I<157F9138 03FFC0020F13E0EC3F8191387E00F002F81370903903F003F0903807E007EB0FC0EB1F80 020013E04914C0017E90C7FC13FE5B485AA21203485AA2380FE07E9038E3FF809038E783 E0391FCE01F09038DC00F813F84848137C5B49137EA2485AA290C7FC15FE5A5AA214015D 5AA214035DA348495A5D140F5D4A5A6C49C7FC127C147C6C485A6C485A6CB45A6C1380D8 01FCC8FC243A76B72A>IIII<133C 137E13FF5AA313FE13FCEA00701300B2120EEA3F80127F12FFA31300127E123C102477A3 19>II< EE01C01603A21607160FA2161F83163FA2167F16FF16EF150116CFED038FA2ED070FA215 0E151E151C1538A203707FA2EDE007A2EC01C014031580EC0700A2140EA25CA25C027FB5 FCA291B6FC9139E00007F849481303A2495A130791C7FC5B130E5BA25B1378137013F0EA 03F8486C4A7EB56C48B512F0A3343C7BBB3E>65 D<0107B612FCEFFF8018C0903B000FF0 001FF04BEB07F81703021F15FC17014B14FEA2023F1400A24B1301A2147F18FC92C71203 18F84A140718F04AEC0FE0EF1FC00101ED3F80EF7F004AEB01FEEE07F849B612E05F9139 F80007F0EE01FC01076E7E177F4AEC3F80A2010F16C0171F5CA2131F173F5CA2133FEF7F 805C1800017F5D4C5A91C7485A5F49140FEE1FE0494A5A00014AB45AB748C7FC16F816C0 37397BB83A>II<0103B612FEEFFFC018F0903B0007F8000FF84BEB03FCEF00FE020F157FF03F 804B141F19C0021F150F19E05D1807143F19F05DA2147FA292C8FCA25C180F5CA2130119 E04A151FA2130319C04A153FA201071780187F4A1600A2010F16FEA24A4A5A60011F1503 4D5A4A5D4D5A013F4B5A173F4A4AC7FC17FC017FEC03F84C5A91C7EA1FC04949B45A007F 90B548C8FCB712F016803C397CB83F>I<0107B8FCA3903A000FF000034BEB007F183E14 1F181E5DA2143FA25D181C147FA29238000380A24A130718004A91C7FC5E13015E4A133E 167E49B512FEA25EECF8000107147C163C4A1338A2010F147818E04A13701701011F16C0 16004A14031880013F150718004A5CA2017F151E173E91C8123C177C4915FC4C5A491407 0001ED7FF0B8FCA25F38397BB838>I<0107B712FEA3903A000FF000074B1300187C021F 153CA25DA2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201015CA24A13 1E163E010314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A2011F14F0A24A 90C8FCA2133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>II<0103 B5D8F80FB512E0A390260007F8C7381FE0004B5DA2020F153F615DA2021F157F96C7FC5D A2023F5D605DA2027F14016092C7FCA24A1403605CA249B7FC60A202FCC712070103150F 605CA20107151F605CA2010F153F605CA2011F157F95C8FC5CA2013F5D5F5CA2017F1401 5F91C7FC491403007FD9FE01B512F8B55BA243397CB83E>I<0103B512F8A390390007F8 005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25CA21303 A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FCA25C25 397CB820>I<0103B500F890387FFFE0A21AC090260007F8C7380FFC004B15E061020F4B C7FC183E4B5C18F0021F4A5A4D5A4BEB0F804DC8FC023F143C5F4B5B4C5A027FEB07C04C C9FCED001E5E4A5BED01FCECFE0315070101497E151FECFC7C4B7E903903FDE07FDAFFC0 7F1580ED003F49488014F84A131F83130F160F4A801607011F81A24A130383133F16014A 80A2017F6E7EA291C8FC494A7F007F01FE011F13FCB55CA243397CB840>75 D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA2 5CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478A201 3F157017F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF0001020713 00B8FCA25E2E397BB834>I<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA202 1F167FF1EFC0141DDA1CFCEC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7 FC0270151CA202F04B5AF0707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED07 01610280140EA20107ED1C0305385B14006F137049160705E05B010EEC01C0A2011E9138 03800F61011CEC0700A2013C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA2 01F04A5B187E00015DD807F816FEB500C09039007FFFFC151E150E4C397AB84A>I<9026 03FFF891B512E0A281D90007923807F8006F6E5A61020F5E81DA0E7F5DA2021E6D130703 3F92C7FC141C82DA3C1F5C70130EEC380FA202786D131E0307141C147082DAF003143C70 133814E0150101016E1378030014705C8201036E13F0604A1480163F010715C1041F5B91 C7FC17E149EC0FE360010E15F31607011E15FF95C8FC011C80A2013C805F133816001378 5F01F8157CEA03FC267FFFE0143CB51538A243397CB83E>II<0107B612F817FF1880903B000F F0003FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15 F817074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD9 07F8C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512 FCA337397BB838>II<0103B612F017FEEFFF 80903B0007F8003FC04BEB0FF01707020FEC03F8EF01FC5DA2021F15FEA25DA2143FEF03 FC5DA2027FEC07F818F092C7120F18E04AEC1FC0EF3F004A14FEEE01F80101EC0FE091B6 128004FCC7FC9138FC003F0103EC0F80834A6D7E8301071403A25C83010F14075F5CA201 1F140FA25CA2133F161F4AECE007A2017F160F180E91C7FC49020F131C007F01FE153CB5 913807F078040313F0CAEAFFE0EF3F80383B7CB83D>I<92383FC00E913901FFF01C0207 13FC91391FC07E3C91393F001F7C027CEB0FF84A130749481303495A4948EB01F0A2495A A2011F15E091C7FCA34915C0A36E90C7FCA2806D7E14FCECFF806D13F015FE6D6D7E6D14 E0010080023F7F14079138007FFC150F15031501A21500A2167C120EA3001E15FC5EA300 3E4A5AA24B5AA2007F4A5A4B5A6D49C7FC6D133ED8F9F013FC39F8FC03F839F07FFFE0D8 E01F138026C003FCC8FC2F3D7ABA2F>I<0007B812E0A25AD9F800EB001F01C049EB07C0 485AD900011403121E001C5C003C17801403123800785C00701607140700F01700485CA2 140FC792C7FC5DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303 A25CA21307A25CA2130FA25CEB3FF0007FB512F8B6FCA2333971B83B>I<003FB539800F FFFEA326007F80C7EA7F8091C8EA3F00173E49153CA2491538A20001167817705BA20003 16F05F5BA2000715015F5BA2000F15035F5BA2001F150794C7FC5BA2003F5D160E5BA200 7F151E161C90C8FCA2163C4815385A16781670A216F04B5A5E1503007E4A5A4BC8FC150E 6C143E6C6C5B15F0390FC003E03907F01FC00001B5C9FC38007FFCEB1FE0373B70B83E> III<91B712F0A25B9239E0001FE092C7EA3FC0D903FCEC7F8002F015004A14FE 16014948495A4A495A4C5A49C75B4C5A010E143F011E4A5A011C4AC7FC4B5A5E90C7485A 15074B5A4B5A4B5A5E157F4BC8FC4A5A4A5A4A5A5D140F4A5A4A5A4A5A4AC712E05C1301 4948130149485C495A494813034A5C013F1407495A49C7FC48484AC7FC48485C5B000715 3E4848147E4848EB01FE4848EB07FC4848133F90B6FCB7FC5E34397AB833>90 DI93 D<1318133813F0EA01C013801203EA0700120E12 0C121C5A1230A212701260A212EFEAFF80A6EA7F00123C0D196FB919>96 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120FEBC001001F 5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F15831680143F15 87007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F000F0222677 A42A>I<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE90 38EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214075A12 7EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B383C03E0 383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380FC1E090 381F0070017E13784913383901F801F83803F003120713E0120FD81FC013F091C7FC485A A2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC0F806C EB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F903803FFC090380FC1 E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0EC1F80 397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003EEB01E0 EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>IIIII<150E153F157FA3157E151C1500ABEC1F80EC7FC0ECF1F0EB01C0 90380380F813071401130F130E131EEB1C03133C013813F0A2EB0007A215E0A2140FA215 C0A2141FA21580A2143FA21500A25CA2147EA214FEA25CA21301A25CA213035C121C387E 07E0A238FE0FC05C49C7FCEAF83EEA787CEA3FF0EA0FC0204883B619>IIIII<147F903803FFC090380FC1F09038 1F00F8017E137C5B4848137E4848133E0007143F5B120F485AA2485A157F127F90C7FCA2 15FF5A4814FEA2140115FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB3F 00147E6C13F8380F83F03803FFC0C648C7FC202677A42A>I<9039078007C090391FE03F F090393CF0787C903938F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01E1 13C15CA2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E01 3F495A6E485A5E6E48C7FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201A2 5BA21203A25B1207B512C0A3293580A42A>II<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F800078 150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA312075BA3120F5B A3121F5BA3123F90C9FC120E212679A423>I<14FE903807FF8090380F83C090383E00E0 4913F00178137001F813F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13F814 FE6C7F6D13807F010F13C01300143F141F140F123E127E00FE1480A348EB1F0012E06C13 3E00705B6C5B381E03E06CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E0F8007121E 121C0038140F131F007815C01270013F131F00F0130000E015805BD8007E133FA201FE14 005B5D120149137EA215FE120349EBFC0EA20201131E161C15F813E0163CD9F003133814 070001ECF07091381EF8F03A00F83C78E090393FF03FC090390FC00F00272679A42D>I< 01F0130ED803FC133FD8071EEB7F80EA0E1F121C123C0038143F49131F0070140FA25BD8 F07E140000E08013FEC6485B150E12015B151E0003141C5BA2153C000714385B5DA35DA2 4A5A140300035C6D48C7FC0001130E3800F83CEB7FF8EB0FC0212679A426>I<01F01507 D803FC903903801F80D8071E903907C03FC0D80E1F130F121C123C0038021F131F49EC80 0F00701607A249133FD8F07E168000E0ED000313FEC64849130718000001147E5B03FE5B 0003160E495BA2171E00070101141C01E05B173C1738A217781770020314F05F00030107 13016D486C485A000190391E7C07802800FC3C3E0FC7FC90393FF81FFE90390FE003F032 2679A437>I<903907E007C090391FF81FF89039787C383C9038F03E703A01E01EE0FE38 03C01F018013C0D8070014FC481480000E1570023F1300001E91C7FC121CA2C75AA2147E A214FEA25CA21301A24A1370A2010314F016E0001C5B007E1401010714C000FEEC038001 0F1307010EEB0F0039781CF81E9038387C3C393FF03FF03907C00FC027267CA427>I<13 F0D803FCEB01C0D8071EEB03E0D80E1F1307121C123C0038140F4914C01270A249131FD8 F07E148012E013FEC648133F160012015B5D0003147E5BA215FE00075C5BA214015DA314 035D14070003130FEBF01F3901F87FE038007FF7EB1FC7EB000F5DA2141F003F5C48133F 92C7FC147E147C007E13FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA03F0233679 A428>I<903903C0038090380FF007D91FF81300496C5A017F130E9038FFFE1E9038F83F FC3901F007F849C65A495B1401C7485A4A5A4AC7FC141E5C5C5C495A495A495A49C8FC13 1E5B49131C5B4848133C48481338491378000714F8390FF801F0391FFF07E0383E1FFFD8 3C0F5B00785CD8700790C7FC38F003FC38E000F021267BA422>I I<001E1338007F13FEEAFF811383A3EB03FC00FE13F8383800F017096AB72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmex10 10 23 /Fq 23 114 df<1430147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B 1203A2485AA3485AA3121F90C7FCA25AA3123EA2127EA6127C12FCB3A2127C127EA6123E A2123FA37EA27F120FA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E0147014301462738226>0 D<12C07E12707E123C7E7EA26C7E6C7EA26C7E7F12 007F1378137CA27FA37FA31480130FA214C0A31307A214E0A6130314F0B3A214E01307A6 14C0A2130FA31480A2131F1400A3133EA35BA2137813F85B12015B485AA2485A48C7FCA2 121E5A12385A5A5A14627C8226>I<1538EC01F8EC07E0EC1F80EC7E005CEB03F85C495A A2495AB3AB131F5CA249C7FC137E5BEA03F8EA07E0EA3F8000FCC8FCA2EA3F80EA07E0EA 03F8C67E137E7F6D7EA280130FB3AB6D7EA26D7E80EB00FC147EEC1F80EC07E0EC01F8EC 00381D62778230>8 D<12E012FCEA3F80EA07E0EA03F8C67E137E7F6D7EA280130FB3AB 6D7EA26D7E80EB00FC147EEC1F80EC07E0EC01F8A2EC07E0EC1F80EC7E005CEB03F85C49 5AA2495AB3AB131F5CA249C7FC137E5BEA03F8EA07E0EA3F8000FCC8FC12E01D62778230 >I<12F0B3B3B2043674811C>12 D<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F 8015005C147E5CA2495A495AA2495AA2495AA2495AA249C7FCA2137EA213FE5B12015BA2 12035BA21207A25B120FA35B121FA45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7F A4120F7FA312077FA21203A27F1201A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7E A26D7E6D7EA2147E80801580EC0FC0EC07E01403EC01F0EC00F8157C153E151E1F947182 32>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA2 6D7EA26D7EA26D7EA280147E147F80A21580141FA215C0A2140F15E0A3140715F0A41403 15F8A5EC01FCA9EC00FEB3A8EC01FCA9EC03F8A515F01407A415E0140FA315C0141FA215 80A2143F1500A25C147E14FE5CA2495AA2495AA2495AA2495AA2495AA249C7FC137EA25B 485A5B1203485A485A5B48C8FC123E5A5A5A1F947D8232>I<161E167EED01FE1507ED0F F8ED3FE0ED7FC0EDFF80913801FE004A5A4A5A5D140F4A5A5D143F5D147F92C7FCA25C5C B3B3B3A313015CA3495AA213075C495AA2495A495A137F49C8FC485A485AEA07F0EA1FE0 485AB4C9FC12FCA2B4FCEA3FC06C7EEA07F0EA03FC6C7E6C7E6D7E133F6D7E6D7EA26D7E 801303A26D7EA3801300B3B3B3A38080A281143F81141F816E7E1407816E7E6E7E913800 FF80ED7FC0ED3FE0ED0FF8ED07FE1501ED007E161E27C675823E>26 D56 D<12F812FE6C7E7F13F0EA3FF86C7E6C7EEA03FF6C7F6C7F6D7E6D7E806D7E130F6D7E80 7F15807F15C07FA2EC7FE0A3EC3FF0A415F8141FB3B3A71D4B737E4A>IIIII80 D<167F923801FFC0923803C0F09238 07803892380F007892381F01FC151E153EA2157E92387C0070170015FCA44A5AA81403A4 5DA41407A94A5AAA4A5AA95DA4143FA492C8FCA7143E147EA4147C123800FE13FC5CA249 5A5CEA7803387007C0383C0F80D80FFEC9FCEA03F82E5C7C7F27>82 D<0078EF078000FCEF0FC0B3B3B3A46C171F007E1880A2007F173F6C1800A26D5E001F17 7E6D16FE6C6C4B5A6D15036C6C4B5A6C6C4B5A6C6C4B5A6C6C6CEC7FC06D6C4A5AD93FF8 010790C7FC6DB4EB3FFE6D90B55A010315F06D5D6D6C1480020F01FCC8FC020113E03A53 7B7F45>I<913801FFE0020F13FC027FEBFF8049B612E04981010F15FC499038003FFED9 3FF8EB07FFD97FC001007F49486E7E4848C8EA1FE048486F7E48486F7E48486F7E491501 48486F7E49167E003F177F90CA7EA2481880007E171FA200FE18C048170FB3B3B3A40078 EF07803A537B7F45>I88 D90 D101 D<1B301B781BF8A2F201F0A2F203E0A2F207C0A2F20F 80A2F21F00A21A3EA262A262A24F5AA24F5AA24F5AA262190FA24FC7FCA2193EA261A261 A24E5AA24E5AA24E5AA24E5AA24EC8FCA2183EA260131001305E13F800014C5A1203D80F FC4B5A121DD838FE4B5A12F0D8407F4B5A12004DC9FC6D7E173E6D7E5F6D7E5FA26D6C49 5AA26D6C495AA26D6C5C1607A26D6C495AA2027F49CAFCA291383F803EA25EEC1FC05EEC 0FE0EDE1F0EC07F1EDF3E0A26EB45AA26E5BA26E90CBFCA25D157E157C15384D64788353 >112 D<1B301B78A21BF8A21BF01A01A21BE01A03A21BC01A07A21B801A0FA21B0062A2 1A1E1A3EA21A3C1A7CA21A781AF8A262A21901A2621903A2621907A262190FA297C7FC61 A2191E193EA2193C197CA2197819F8A2611801A2611803A261A21807A261180FA296C8FC 60A2181E183EA2183C187C131001301678017016F813F860000116011203486C5E000F16 03121DD838FE5E00701607126000C05FEA407F0000160FA26D6C92C9FC5FA2171E6D6C14 3EA2173C6D6C147CA2177817F86D7E5F16016D7E5F1603A26D6C5C1607A26D6C5C160FA2 94CAFC027F5BA2161EEC3F80163EA2163C91381FC07CA2167891380FE0F8A25E15E1EC07 F15E15F3EC03FB5E15FFA26E5BA36E90CBFCA35D157EA2157C153C15384D96788353>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbsy10 10 1 /Fr 1 2 df1 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx12 12 50 /Fs 50 124 df12 D45 DI48 DIII<163FA25E 5E5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14 F8EB01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E 5A12FCB91280A5C8000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01 FFEB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01 C714C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0 C8FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0 123E003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F 01FCC7FC010113C02D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F80 90390FFC001FD93FF014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0 EE7F80EE1E00003F92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039 F9F01FFC9039FBC007FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA412 7FA5123FA217F07F121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C90 38E01FFC6DB55A011F5C010714C0010191C7FC9038003FF02D427BC038>I56 D65 D68 DI71 DII75 D77 D<923807FFC092B512FE0207ECFFC0021F15F091267FFE0013FC902601FFF0 EB1FFF01070180010313C04990C76C7FD91FFC6E6C7E49486F7E49486F7E01FF8348496F 7E48496F1380A248496F13C0A24890C96C13E0A24819F04982003F19F8A3007F19FC4917 7FA400FF19FEAD007F19FC6D17FFA3003F19F8A26D5E6C19F0A26E5D6C19E0A26C6D4B13 C06C19806E5D6C6D4B13006C6D4B5A6D6C4B5A6D6C4B5A6D6C4A5B6D01C001075B6D01F0 011F5B010101FE90B5C7FC6D90B65A023F15F8020715C002004AC8FC030713C047467AC4 54>79 DI83 D<003FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607A2 007E1803007C1801A300781800A400F819F8481978A5C81700B3B3A20107B8FCA545437C C24E>I86 DI<9038 01FFE0011F13FE017F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7F A36F7F6C5A6C5AEA00F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0 000F1380381FFE00485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC0 3A1FFF80FC7F0007EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 DIIIIIII<137C48 B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FCAAEB7FC0EA7FFF A512037EB3AFB6FCA518467CC520>I107 DI<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8F001B512E002816E48 80913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F009026FF3E007F6C019E6D 013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B60003B512FEA5572D7C AC5E>I<90397F8007FEB590383FFF8092B512E0028114F8913987F03FFC91388F801F00 0390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B512FEA5372D7CAC3E> II<90397FC00FF8B590B57E 02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF6C496D13804A15C04A 6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B6E4913 806E4913006E495A9139DFC07FFC02CFB512F002C314C002C091C7FCED1FF092C9FCADB6 7EA536407DAC3E>II<90387F807FB53881FFE0028313F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14 F8A214F0ED0FFC9138E007F8ED01E092C7FCA35CB3A5B612E0A5272D7DAC2E>I<90391F FC038090B51287000314FF120F381FF003383FC00049133F48C7121F127E00FE140FA215 077EA27F01E090C7FC13FE387FFFF014FF6C14C015F06C14FC6C800003806C15806C7E01 0F14C0EB003F020313E0140000F0143FA26C141F150FA27EA26C15C06C141FA26DEB3F80 01E0EB7F009038F803FE90B55A00FC5CD8F03F13E026E007FEC7FC232F7CAD2C>IIIIIII123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr5 5 6 /Ft 6 53 df<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D48 D<1360EA01E0120F12FF12F1 1201B3A3387FFF80A2111C7B9B1C>IIII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmsy7 7 18 /Fu 18 108 df0 D<1238127C12FEA3127C123807077A9114>I< 0060140600F0140E0078141E6C143C6C14786C14F039078001E03903C003C03901E00780 3900F00F00EB781E6D5A6D5A6D5A6D5A6D5A497E497EEB1E78497E497E497E3901E00780 3903C003C039078001E048C712F0001E147848143C48141E48140E006014061F1F769D34 >I<1338A50060130C00F8133E00FC137E00FE13FE383FBBF83807FFC000011300EA007C 48B4FC000713C0383FBBF838FE38FE00FC137E00F8133E0060130C00001300A517197B9A 22>I<1406140EB3B812E0A3C7000EC8FCB1B812E0A32B2B7CA834>6 DI<176017F01770 A217781738173C171C171E83717E717E717EEF00F8BAFC19801900CB12F8EF01E04D5A4D 5A4DC7FC171E171C173C173817781770A217F01760391F7C9D42>33 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA1F80A21300A25A123EA35AA312 7812F8A25A12100D1E7D9F13>48 D<017F157F2601FFE0903803FFC0000701F890380FF1 F0260F83FC90381F0038261E00FF013C7F001890263F8078130C4890261FC0E07F007090 260FE1C07F0060EB07E3913803F780486DB4C7EA01806E5A157E157F81824B7E0060DAF7 E0EB0300913801E3F0DBC3F85B6C90260381FC13066C90260F00FE5B001C011E90387F80 3C6C017C90381FE0F82607C7F86DB45A2601FFE0010313C06C6CC86CC7FC391B7C9942> I<49B5FC130F133F01FFC7FCEA01F8EA03E0EA078048C8FC121E121C123C123812781270 A212F05AA2B7FCA300E0C8FCA27E1270A212781238123C121C121E7E6C7EEA03E0EA01F8 6CB4FC013FB5FC130F130120277AA12D>I<150EA2151E151C153C1578157015F015E014 0115C0140315801407EC0F00140E141E141C143C14381478147014F0495A5C13035C1307 91C7FC5B131E131C133C13381378137013F05B1201485A5B120790C8FC5A120E121E121C 123C5A127012F05A12601F3576A800>54 D<161E163EA2167E16FEA2150116BE1503163E 15071506150E151CA215381530ED703F156015E04A487EA2EC0380EC0700A2140E5C835C 027FB5FCA291B6FC903901C0000F13034A80D84007C7FCEA600ED8F01E1407D8FC7C81B4 5A49EDF78049913803FE006C485D6C48EC01F0000ECBFC312D7DA935>65 D69 D<49B6FC011F14FE017F14FC9039F8007800D803C05B38 078001D80F005B481303001E5CC712075D140FA292C7FC5CA3141E143EA3143C147CA214 7814F8A25C1301A25C13035C010714E0EC8003D90F0013C0011EEB0780003FB5EAFE0048 5CB612E0282880A726>73 D81 D83 D<12E0B3B3B3A5033B78AB14>106 D<38C00180EAE003B3B3B3A3EAC001113B78AB22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmr7 7 23 /Fv 23 127 df22 D<1306130C13181330136013E0EA01C0EA03 80A2EA07005A120E121EA2121C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E12 0F7EEA0380A2EA01C0EA00E0136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013E0120013F0A213701378A3133CA5133E13 1EAB133E133CA51378A3137013F0A213E0120113C0EA0380A2EA0700120E120C5A5A5A5A 0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8FCB3A22B2B7DA333>43 D48 D<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215267BA521 >I<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4127CC7FC 15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA0180390300 030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF000313E0380F01F8381C007C 0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB07E03801FF8091C7FC38 0001E06D7E147C80143F801580A21238127C12FEA21500485B0078133E00705B6C5B381F 01F03807FFC0C690C7FC19277DA521>I<1438A2147814F81301A2130313071306130C13 1C131813301370136013C012011380EA03005A120E120C121C5A12305A12E0B612E0A2C7 EAF800A7497E90383FFFE0A21B277EA621>I<0018130C001F137CEBFFF85C5C1480D819 FCC7FC0018C8FCA7137F3819FFE0381F81F0381E0078001C7F0018133EC7FC80A21580A2 1230127C12FCA3150012F00060133E127000305B001C5B380F03E03803FFC0C648C7FC19 277DA521>I I<1230123C003FB512E0A215C0481480A239700007000060130E140C48131C5C5CC75A5C 1301495AA249C7FC5B130E131EA3133E133CA2137CA413FCA813781B287DA621>I<137F 3803FFE0380781F8380E007C48131E5A801278A3127C007E131EEA3F80EBE03C6C6C5A38 0FFCF03807FFC06C5BC613E0487F38079FFC380F07FEEA1E0348C67E48133FEC1F804813 0FA21407A315001278140E6C5B6C5B380F80F03803FFE0C66CC7FC19277DA521>I<137F 3801FFC03807C1E0380F0070001E1378003E7F003C133E007C131EA200FC131FA41580A4 007C133FA2123C003E137F121E380F01DF3807FF9F3801FE1FD8001013001300A2143E12 3C007E133CA25C5C007C5B383003C0381C0780D80FFFC7FCEA03F819277DA521>I61 D91 D93 D100 D<120EEA3F80A5EA0E00C7FCA7EA078012FFA2121F120FB3121FEAFFF8A20D287EA713> 105 D111 D<3803F840380FFEC0EA3C07EA7803EA7001EAF000A37E6C1300EA 7FC013FC6CB4FC6C1380000713C0C613E0130738C003F0130113007EA26C13E0130100F8 13C038EE078038C7FF00EA81FC141C7E9A1A>115 D<13C0A41201A312031207120F121F B512E0A23807C000AC1430A73803E060A23801F0C03800FF80EB3F0014257FA31A>I<38 0F8010381FF038383FFFF04813E038E07FC038400F8015067BA621>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmmi7 7 52 /Fw 52 122 df11 D<137E48B4EB0180000713C0489038E00300481406383E01F0393800700C4813380060EB 181848131CEC0C305AC75B14065DA3EC0780A292C7FCA31406A3140EA2140C141CA45CA3 1430A2142021267E9923>13 DII21 D<137001F81338157CA248485BA44848485AA44848485AA44848485AEDC180A3001F9038 0F8300A2141F9038C03786393FE0E7CC9038FFC3FC393E7F00F090C9FC5AA45AA45A5A21 267D9928>II<497EA414FF01071380131F90387C7F0049C7FC485A485A 5B1207A2485AA46C7EA23803EFF06CB47E7E3803DFF0D80780C7FC000EC8FC121E5A1238 1278127012F0A37E7E7EEA7FC013F8EA3FFE380FFFC0000313F8C67F131FEB03FEEB007E 143E141CEB203CEB7838EB3FF0EB07C019347EA71E>I<48B61280000715C04815804815 00263C0C06C7FC127012C0EB1C0EEA0018A21338A2EB701EA313F013E01201141F120313 C0000780A2380F800FA26C486CC7FC221A7D9827>I<14FCEB03FF903807878090381E03 C0EB3C01017813E0A213F0000114F013E01203A23907C003E0A4390F8007C0A21580EC0F 00EA1F00141E6D5A6D5A383EE1F0EB7FC0011FC7FC90C8FC5AA45AA45A5A1C267D9922> I<48B512F8000714FC4814F84814F0D83C07C7FC1270EAC006130E1200A3131E131CA213 3CA35BA313F8A3485AA26C5A1E1A7D981F>28 D<16E0000214010006EC03F0120E000C14 01481400A24815701660481330147014F04815C0A2495AED018014C0ED03005DD8E00313 0E0107131E39F01FE07CB65AD87FFC5B6C485B391FE07FC0260F803FC7FC241B7E992A> 33 DI<1238127C12FE12FFA2127F123B1203A31206A3120C121812381270 122008127A8614>59 D61 D64 D<4B7E1503150782150F151FA2153FA2156F15CF82EC0187140315071406140E140C0218 7FA2EC30031460A214C013011480D903007F91B5FC5B90380C0001A25B13380130805B01 E013005B12011203000F4A7ED8FFF890381FFFE0A22B2A7DA932>I<013FB612FCA29039 01FC00014AEB007C173C0103153817185CA21307A24A13C0A2010F010113005E14C01503 011F130F91B5C7FCA2EC800F013F7F15061400A249010E13E0030C13C0017E90C7FC1601 01FEEC0380A249EC0700A20001150E161E495C16FC0003EC07F8B7FC5E2E287DA731>69 D<903B3FFFF01FFFF8A2D901FCC7EAFE004A5CA2010314015F5CA2010714035F5CA2010F 14075F5CA2011F140F91B65AA2913880000F013F141F5F91C7FCA249143F94C7FC137EA2 01FE5C167E5BA2000115FE5E5BA200031401B539C07FFFE0A235287DA736>72 D<91387FFFE0A2913800FE005DA214015DA314035DA314075DA3140F5DA3141F5DA3143F 92C7FCA35C001E137E127FA214FE00FE5B387E01F8387803F0387007E0383C1FC0D81FFF C8FCEA03F823297CA725>74 D<90263FFFF0EB7FF8A2D901FCC7EA1FC04AEC1E005F0103 15704C5A4AEB03804CC7FC0107141C5E4A13E04B5A010FEB0780030EC8FC4A5A157C011F 13FE14C3EC877F149E90393FB83F8014F09138C01FC0148049486C7EA2017E6D7EA201FE 6D7EA2496D7EA200016E7EA249147FA2000382B539C007FFF8A235287DA738>I<90383F FFF8A2D901FCC7FC5CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C8FC A249141C1618137E163801FE1430167049146016E000011401ED03C0491307ED0F800003 147FB7FC160026287DA72E>III<013FB512E016FC903901FC007F4AEB0F80EE07C001 0315E016035C17F01307EE07E05CA2010FEC0FC017804AEB1F00163E011F14F8ED07F091 B51280A290393F800FE0ED03F002007F15015BA2137EA201FE1303A2495CA20001160817 184914E017380003EDF070B5D8C00113E0923800FFC0C9EA3F002D297DA732>82 D<91381FE0089138FFFC18903903E01E3890390780077090390E0003F049130149130001 7814E0137013F0A2000115C0A216007F7F6CB47E14F86DB47E6D13F06D7F01077F01007F 1407EC00FF153F81A3001880A20038141E12300038141C153C00781438007C5C007E5C00 77EB03C026E3E00FC7FC38C0FFFE38801FF0252A7CA829>I<000FB712E05A9039800FE0 07D81E009038C001C05A0038011F1300123000705C00601501023F148012E0481400A2C7 4890C7FCA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131F00 1FB57EA22B287DA727>I<903B3FFFE00FFFC0A2010190390001FC006D4814F017C0027F 495A4CC7FC6E130E6F5A021F5B6F5A5E91380FE1C0EDE380DA07F7C8FC15FE6E5A5D6E7E A2811403EC077F140E4A7E02187FEC301F02607F14C049486C7EEB030001066D7E5B0138 6D7E5B01F06D7E485AD80FF0497ED8FFFC90381FFFE0A232287DA736>88 DI97 DII<15F8141FA2EC01F0A21403A215E0A21407 A215C0A2140FEB1F8F90387FCF80EBF0EF3803C03FEA0780390F001F00A2001E5B123E00 3C133E127C147E5A147CA214FC5AECF830A3903801F060A2EA7803010E13C0393C1CF980 381FF07F3907C01E001D297CA723>IIII<13 3EEA07FEA2EA007CA213FCA25BA21201A25BA2120314FCEBE3FF9038EF0780D807FC13C0 EBF00313E0A2EA0FC014071380A2121FEC0F801300A248EB1F00A2003E1406143E127EEC 7C0C127C151800FCEB3C30157048EB1FE00070EB0F801F297CA727>I<130E131F5BA213 3E131C90C7FCA7EA03E0487EEA0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485A A2EBC180EA0F81A2381F0300A213066C5A131CEA07F06C5A11287DA617>I<1407EC0F80 141FA21500140E91C7FCA7EB03E0EB07F8EB0C3C1318EB303E136013C0A248485AA2C7FC A25CA4495AA4495AA4495AA4495AA21238D87C1FC7FC12FC133E485AEA70F8EA7FE0EA1F 80193380A61B>I<133EEA07FEA2EA007CA213FCA25BA21201A25BA21203EC07809038E0 1FC0EC38600007EB61E014C3EBC187EBC307D80FC613C09038CC038001B8C7FC13E0487E 13FEEB3F80EB0FC0486C7E1303003E1460A2127EECC0C0127CECC18012FC903801E30038 F800FE0070137C1B297CA723>I<137CEA0FFCA2EA00F8A21201A213F0A21203A213E0A2 1207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2EA7C18A3EAF830A21320 EA786013C0EA3F80EA0F000E297EA715>I<3B07801FC007E03B0FE07FF01FF83B18F0E0 F8783C3B30F1807CE03E903AFB007D801ED860FEEB3F005B49133E00C14A133E5B1201A2 4848495BA35F4848485A1830EE01F0A23C0F8003E003E060A218C0933801E180271F0007 C013E3933800FF00000E6D48137C341B7D993B>I<3907801FC0390FE07FF03918F0E0F8 3930F1807CEBFB00D860FE133C5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03 E0A23A0F8007C0C0A2EDC180913803C300D81F0013C7EC01FE000EEB00F8231B7D9929> II<9038F007C03901FC1FF039031E78780006EBE03C90381FC01C000CEB801E 14005B0018141F133E1200137E153E137CA213FC157C5B1578000114F0A2EC01E0EC03C0 3903FC07809038FE1F00EBE7FCEBE1F0D807E0C7FCA25BA2120FA25B121FEAFFF8A22025 809922>II<3807803E390FE0FF803818 F3C13930F703C0EBFE073860FC0F13F8158039C1F0070091C7FC1201A2485AA4485AA448 5AA448C8FCA2120E1A1B7D991F>II<131C133EA25BA4 5BA4485AB512E0A23801F000485AA4485AA4485AA448C7FC1460A214C0123EEB0180EB03 00EA1E06EA1F1CEA0FF8EA03E013267EA419>II<90387C03C039 01FF0FF03907079C30390E03B078000CEBF0F8001813E1123015F0396007C0E015001200 A2495AA449C7FC15301238007C1460EAFC3E15C0EAF87E39F06F03803970C70700383F83 FE381F01F81D1B7D9926>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx msbm10 10 6 /Fx 6 124 df67 D78 D<007FB612E0B712FE6C6F 7E2703C01E0313E0000190393C00F3F00238EB70F8EE783CEE381E83EE3C07161C188017 03A617071800EE3C0FEE380E173EEE78FCEEF7F892380FFFE0023FB5128004FCC7FC16B8 913838F03CED701CED781EED380EED3C0FED1C07031E7FED0E03030F7FED0701EE81E0ED 0380707E030113701778EEE0380300133C707EEE700EEE780F9338380780EE3C03041C13 C093381E01E00003013C90380E00F0007FB539F00FFFFEB67F6C8137397DB836>82 D<007FB812C0B9FCA23BE1FE38071FE1D8E3F0EC03F1D8E7C0EC00F9D8EF00153D00FE16 1F48160F481607A2481603A2481601A400601600C71600B3B102F813C0011FB6FC5B7F32 397DB838>84 D<0007B712FC5AA23B0E1FF003C038903A3F0007807801FC4A5AD80FF049 5B49EB0E01D81F80011E5BED3C0390C738380780001E027890C7FCED700FEDF00E001C90 3801E01E4B5A02031338C7EB80780207137091380F00F091380E01E0021E5BEC1C03023C 5BEC3807027890C8FC4A5AECE01E0101131CECC03C0103133890380780784A5A495BEB0E 01011E49EB0180D93C0314039038380780017890C7FCD9700F1407EBF00E3801E01E4948 EC0F0000031338D980785C00071370D900F05C48495CD81E0115F7261C03C0EB01E7003C 49495AD83807EC0F8E007890C7EA3F0E4848EB01FEB812FEA331397DB83E>90 D<02F815FCD907FCEC01FE011F1507013F150F017FED3FF801FFED7FF0D9F81C903801FF C0D801C04A1380D9801890380FFE004C5AC7EC7FF00238495ADA30035B5DDA701F5B9126 603FFBC7FCEDFFE702E113C602C7130E903901CFFE0CECBFF8903903FFF01C49EBC01815 8090390FFE003849481330D97FF01403495A00030180EB70074890C7133ED81FFCEC7FFE 48485DD8FFE05D495D90C813C0007E033EC7FC37247CA337>123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmsy10 10 49 /Fy 49 115 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB 01F86C6CEB03F06C6CEB07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380F C1F8903807E3F0903803F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090 380FC1F890381F80FC90383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F0 4848EB01F84848EB00FC48C8127E007E153F48151F48150F00601506282874A841>II<15301578 B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A3007FB812F8B912FCA26C17F836367B B641>6 D<007FB812F8B912FCA26C17F8C80078C8FCB3A3007FB812F8B912FCA26C17F8 C80078C8FCB3A6153036367BA841>I9 D14 D<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F8CCFCAE007FB8 12F8B912FCA26C17F836287BA841>17 D20 D<126012F812FEEA7F80EA3FE0EA0FF8 EA03FEC66C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3F E0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FC0171FEF7F809338 01FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC 7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CC FCAE007FB81280B912C0A26C1780324479B441>I24 D<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FECA FCEA01F8485A485A485A5B48CBFCA2123EA25AA2127812F8A25AA87EA21278127CA27EA2 7EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F15803232 79AD41>26 D<007FB512FCB712C016F06C15FCC8EA07FE9238007F80EE1FC0EE07E0707E 707E707E177C83A283A2EF0F80A2170718C0A21703A81707A21880170FA2EF1F00A2173E A25F17FC4C5A4C5A4C5AEE1FC0EE7F80DB07FEC7FC007FB65AB712F016C06C02FCC8FC32 3279AD41>I<05041402051E140F057E143FDC01FE14FF4C48EB01FEDC0FF0EB07F8DC3F C0EB1FE04CC7EA3F80DB01FEECFF00DB07F8EB03FCDB0FE0EB07F0DB3FC0EB1FE003FFC7 EA7F80DA01FC02FEC7FCDA07F8EB03FCDA1FE0EB0FF0DA3F80EB1FC002FFC7EA7F80D903 FCD901FEC8FCD90FF0EB07F84948495AD97F80EB3FC0D801FEC7B4C9FCD803F8EB01FCD8 0FF0EB07F8D83FC0EB1FE048C7EA3F8000FE4ACAFCA2007F6E7ED83FC0EB1FE0D80FF0EB 07F8D803F8EB01FCD801FE6DB4FC26007F80EB3FC0D91FE0EB0FF06D6C6D7ED903FCEB01 FED900FF9038007F80DA3F80EB1FC0DA1FE0EB0FF0DA07F8EB03FCDA01FCEB00FE6EB4EC 7F80DB3FC0EB1FE0DB0FE0EB07F0DB07F8EB03FCDB01FEEB00FFDB007FEC3F80DC3FC0EB 1FE0DC0FF0EB07F8DC03FCEB01FE706CEB00FFDC007E143F051E140F48377BB053>I<18 1EA4181F84A285180785727EA2727E727E85197E85F11F80F10FC0F107F0007FBA12FCBC FCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A61180F96C7FCA2 60181EA4482C7BAA53>33 D<173C173F06C01303DD1FF8131F0507B6FC05015CEF003F18 0795380001FC19031907F10FBCF11F3C193E197C19F8F001F0F003E0F007C0F00F80F01F 00063E7F60604D48131F4D487F4D4814804D4813074DC713C0053EEC03E05F4DEC01C04C 4891C7FC4C5A4C5A4C5A4CCAFC163E5E5E4B5A4B5A4B5A4B5A4BCBFC153E5D5D4A5A4A5A 4A5A4A5A4ACCFC143E5C5C495A495A495A495A49CDFC133E5B5B485A485A485A485A48CE FC123E5A5A5A12604B4A7BBB53>37 D 49 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F8001FEC8FCEA01F8485A485A 485A5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA216FE00F0C9FCA27EA21278 127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF06DB512FE010314FF130002 1F13FE283279AD37>I<387FFFF8B6FC15C06C14F0C7EA0FF8EC01FEEC007FED1F80ED0F C0ED07E0ED03F01501ED00F8A2167CA2163EA2161E161FA2160FA2007FB7FCB8FCA27EC9 120FA2161FA2161E163EA2167CA216F8A2ED01F01503ED07E0ED0FC0ED1F80ED7F00EC01 FEEC0FF8007FB55AB612C092C7FC6C13F8283279AD37>I54 D<126012F0AD12FCA412F0AD126006207BA400>I<0060161800F0163C6C16 7CA200781678007C16F8A2003C16F0003E1501A26CED03E0A26C16C06D1407A200071680 6D140FA26C6CEC1F00A26CB612FEA36C5D01F8C7127CA2017C5CA2013C5C013E1301A201 1E5C011F1303A26D6C485AA201075CECC00FA2010391C7FC6E5AA2903801F03EA2010013 3CECF87CA2EC7878EC7CF8A2EC3FF0A26E5AA36E5AA36E5A6EC8FC2E3C80B92F>I<007F B612F0B712F8A27EC91278B3A5003FB612F85AA27EC91278B3A5007FB612F8B7FCA26C15 F0253A7CB92E>I<18F017011707A3170FA2171F60173F1737177F176F17EF17CF04017F 178F1603170FEE0707160EA2161C161816381630167016E0A2ED01C016801503ED0700A2 150E5DA25D157815705D02018103CFB5FCEC03BF4AB6FCA2020EC71203141E5C14380278 8100205B386001E0EAF0036C4848140126FE1F8081B5C8FC190C49EEFF3C496F13F06C48 17E06C4817806C48EE7E00D8078093C7FC3E407DBB42>65 D<0203B512F0027F14FF49B7 12E0010F16F890273FC3F00713FED978039038007FFF2601E007020F1380D803C0030313 C0D80780030013E0000F177FD81F00EE3FF048EF1FF8003E4A140F5A0078EF07FC00C001 0F1503C7FCA24B1401A3141F5DA3023F16F8A292C8FCF003F0A25C027EED07E0A219C04A 150F1980F01F00495A183E6049481578604D5A49484A5A4D5A050EC7FC4948143C5FEE01 E04948EB07C0043FC8FC91380001FC49EB3FF049B5128048B500FCC9FC4814E04801FCCA FC3E397FB840>68 DI72 D<92B612FC021F15F891B712F0010316C090270FF0003CC7FC013EC7127C01785C491301 00015D0003140348485C49130790C7FCC8485AA34B5AA34BC8FCA35D157EA315FE5DA314 015DA34A5AA314075DA34A5AA25D141F92C9FC4A1406023E141C027E147C027C5C4A495A 4A5C4948495A000FB7C7FC003F5D4815F0B712C0363982B82D>I76 D78 D<0370EBFF80912601E00713E0912603 C01F13F891260F007F7F021E9038F03FFE913A7803C00FFF9139F0078003494848486C13 80902603C01E7F902607803EEC7FC049485A011E49143F013E17E0494848141FEBF8035D 2601F007150F00035CEBE00F00075CD9C01EC8FC000F131C49C9FC121FA248CA13C0A348 171F1980127EA2183F00FE1800A2183E187E187C18FC6017016C5F4D5A6017076C6C4B5A 4DC7FC171E6D5D6C6C5D5F6D4A5A6C6CEC03806C6C020FC8FC01FF143E6C01C013F86C90 38F807E06C90B512806C6C49C9FC011F13F0010313803B3D7BBA42>I<923801FFC0031F 13F8037F13FE0203B6FC91260FE01F138091261E000313C00278010013E04A147FD903C0 EC3FF04948141F49C8EA0FF8131E491507137C49ED03FC485AA2485A48481501A2120F48 5AA290C9FC5AA24817F8127EA2170312FE18F0A3EF07E0A26C17C0170F18806DED1F0012 7F6D153E6D5D6C6C130F01FC013E5B3B1FFF01F801F06CD9FFE05B6C91388003C0000149 48485A26007FE049C7FC90C8121E163816F0ED03E0ED0780033EC8FCEC0FFC0003B500E0 140E000F0280143E4801FCC8127C48D9FF8014FC000102F014F8D8000F01FEEB01F00101 D9FFC013E0D9003F9038FC03C0020790B5120002005C031F13F8030113C0374577BA44> 81 D83 D<1A801907F10F00023FB712FE49B85A01 0F17F0013F17C0494CC7FC2801E00003F0C9FC48481307485A120F48C7485A5A5AA200FE 4A5A5A12F01280C8485AA44BCAFCA415FEA44A5AA44A5AA44A5AA4140F5DA35D141FA25D 143FA292CBFC5CA2147E14FE5CA2495A5C495A5C0102CCFC41427DBB2D>II<922601FFC01330031F9038FF01E0037FECFFC04AB71280020716 0091381F003F023C010013BE027CEC007C4A5D01015E49484A5A4A4A5A49484A5A91C812 0F01044BC7FC90C9123E5F17785F4C5A4C5A4C5A4CC8FC161E4AB512E0020714F8839138 0001E3923803C0F892380780E04BC9FC151E5D5D5D4A5A4A5A4A5A4ACAFC143C4A150C4A 153C494815FC49484A5A4948140349C85B131E494B5A495ED801E04B5A2603DFFF92C7FC 48B6EAC01E48EDFFFC4816F0485ED87000158000C09026003FFCC8FC3C397DB83C>90 D<0060161800F0163CB3B26C167CA2007C16F8A26CED01F0003F15036C6CEC07E06C6CEC 0FC0D807F0EC3F80D803FE903801FF003A00FFC00FFC6DB55A011F14E0010391C7FC9038 007FF82E347CB137>II<14034A7E4A7EA24A7EA34A7EA2EC7CF8A2EC F87CA2ECF03C0101133EA249487EA249486C7EA249486C7EA2EC00034980A2013E6D7EA2 496D7EA20178147801F8147CA2484880A2484880A24848EC0F80A2491407000F16C0A248 C8EA03E0A2003EED01F0A2003C1500007C16F8A248167CA248163C006016182E347CB137 >94 D102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80 131F80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E00495A495A495A 5C130F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D537ABD2A>I<14 C0EB01E01303A214C01307A21480130FA2EB1F00A2131E133EA25BA2137813F8A2485AA2 5B1203A25B1207A2485AA290C7FC5AA2123EA2123C127CA2127812F8A41278127CA2123C 123EA27EA27E7FA26C7EA212037FA212017FA26C7EA21378137CA27FA2131E131FA2EB0F 80A2130714C0A2130314E0A21301EB00C0135278BD20>I<126012F07EA21278127CA212 3C123EA27EA27E7FA26C7EA212037FA26C7EA212007FA21378137CA27FA2131E131FA2EB 0F80A2130714C0A2130314E0A414C01307A21480130FA2EB1F00A2131E133EA25BA21378 13F8A25B1201A2485AA25B1207A2485AA290C7FC5AA2123EA2123C127CA2127812F8A25A 126013527CBD20>I<126012F0B3B3B3B3A91260045377BD17>I<0070131C00F0131EB3B3 B3B3A80070131C175277BD2A>I<126012F07EA21278127CA2123C123EA2121E121FA27E 7FA212077FA212037FA212017FA212007FA21378137CA2133C133EA2131E131FA27F80A2 130780A26D7EA2130180A2130080A21478147CA2143C143EA2141E141FA2801580A21407 15C0A2140315E0A2140115F0A2140015F8A21578157CA2153C153EA2151E150C1F537BBD 2A>110 D 112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmmi10 10 79 /Fz 79 123 df11 DII<1403EC3FF8 91387FFF80D901E313C014800103133F9138001F80ED070092C7FC80A280A28080130180 80130080147F81143F8149B47E130790380F8FF0EB3E0F496C7E13F83801F003D803E07F 1207380FC0011380121FEA3F0014005A127EA212FE5D481301A35DA24813035D6C13075D 127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0D8003FC8FC223D7DBB25>II17 DI<1338017E14F001FEEB07FC151F49133B15F30001EB01C391380383F89039 F80700E0020E90C7FC00035B1478EBF0E0EBF1C03807F78001FEC9FCEBFFE014FE390FE1 FFC09038E01FE09038C007F06E7E001F6D7EA2D98000EB0180A2003F1503020114000100 13F8A2485D1606007E0100130E160C00FE151CED7C3848EC1FF00038EC07C029267CA430 >20 D<133F14C0EB07F06D7E801301A26D7EA3147FA36E7EA36E7EA36E7EA36E7EA36E7E A36E7EA26E7EA214014A7E5C4A7E91381E3F80143C14784A6C7E1301EB03E049486C7EEB 0F80EB1F00496D7E137E5B48486D7E485A485A000F6E7E485A485A48C87E12FE167F4816 800070151F293B7CB930>II<017E1438D83FFE147E16FEA2D801FC14FC12000001 140116F85BED03F0120315074914E0150F000715C0ED1F805BED3F00000F147EA2495B4A 5A001F495A5D49485A4A5A003F49C7FC143EEB00F8495A48485AEB0F80D87E3EC8FC13F8 EAFFE0138000F8C9FC27257CA429>I<1406A6913807FFC04A13E091383F80609138FDFF E0903903F87F804948C7FC495A495A495A137F91C8FC5B5B1201A25BA512007F137E9038 3F3FF090381FFFFC90380FC01C90381FFFF890383C7FE001F0C8FC485A485A485AA248C9 FC121EA25AA2127C1278A312F87EA2127E127F7FEA3FE013FC6CB4FC6C13E06C13F80001 13FF6C6C13C0010F13F001037FEB007F140F14031400A4010C5BEB0E0190380783E09038 01FF80D9007EC7FC234B7EB924>I<013FB612E090B712F05A120717E0270F807006C7FC 391E00600E48140C003813E04813C048141CEAC0011200148001035BA213071400A25B15 78011E137CA3133E133C137C157E13FC5B1201157F1203497FA3D801C0131C2C257EA32F >I<15FE913803FF8091380F83E091383E01F091387C00F85C494813FC0103147C494813 7E5C130F495AA249C7FC16FE5B137EA2150113FE4914FCA20001140316F85BED07F01203 ED0FE04914C0151F000715806DEB3F00157E6D5B390FEE01F09038E707E09038C3FF80D9 C0FCC7FC001F90C8FCA25BA2123FA290C9FCA25AA2127EA212FEA25AA2127027377EA42B >I<013FB512FE90B7FC5A5A4815FE260F801CC7FCEA1E005A00385B5A5A481378C7FC14 7014F0A4495AA31303A3495AA3130FA25C131FA3133FA291C8FC131E28257EA324>28 D<1503A35DA21506A2150EA2150CA2151CA21518A21538A21530A21570A2EC07FE91383F FFC0903901FCE3F0903907E0E0F890391F80C03ED93E007FEB7C01D801F8EC0F80D803F0 018013C0D807E014071403D80FC015E0D81F801300A248485AA2007E1306A2020E130F12 FE48010C14C0A2021CEB1F80A20218EB3F00A20238137E007C5D1430007E4A5A003E9038 7003F06CEC07C09138600F80D80F80013FC7FC3903E0E0FC3901F8E7F039007FFF80D90F FCC8FCEB01C0A25CA21303A291C9FCA25BA21306A2130EA2130CA22B4B7CB931>30 DI<160C161C1618A316381630 A316701660A316E05EA315015EA301F80103130FD803FE9138001F80D8070F153F000E01 8015C0001C5C001814060038161F0030160FD8701F010E13070060140C1703D8E03F1680 00C0EB001C491318EA007E180001FE13384913305F000116064913700360130E5F000316 184901E013384B133017705F0201495AD801F849485A4CC7FC160E2600FC035B017EEB00 78013FEB01E090390FE30F80902603FFFEC8FC9038003FF00206C9FCA2140E140CA3141C 1418A314381430A314701460324B7EB936>I<171E01C0153F00015E491680120390C9FC 48163F000E161F000C160F121C0018160718001238003014C014010070010314061260A2 170E00E04948130C5A171C92C71218173817786C010E1470021F14F04A495A6C49495A91 38FF800F3BF801F7C01F80D8FE0F6DB4C7FC3A7FFFE7FFFE02C75B6C01835B02035B260F FC0013C0D803F0013FC8FC31267FA434>II39 D<181EA3181F8485180785727EA2727E851800197C85193FF11F80F10FC0F107F0 F103F8007FBA12FCBCFCA27E48187BAA53>42 D<121C127FEAFF80A5EA7F00121C090979 8817>58 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A120612 0E5A5A5A12600A19798817>II<150C151E153EA2153C157CA2157815F8A215F01401A215E01403A215C01407 A21580140FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA2 5C130FA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A2 5B120FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD2A>I<1260 12FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1F F0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF93 38007F80EF1FC0A2EF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED 7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA 3FF0EA7FC048CBFC12FC1270323279AD41>I64 D<1760177017F01601A21603A21607160FA24C7EA216331673166316C3A2ED0183A2ED03 03150683150C160115181530A21560A215C014011580DA03007FA202061300140E140C5C 021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25BA25B13385B01F01680487E000716 FFB56C013F13FF5EA2383C7DBB3E>I<0103B77E4916F018FC903B0007F80003FE4BEB00 FFF07F80020FED3FC0181F4B15E0A2141FA25DA2143F19C04B143F1980027F157F190092 C812FE4D5A4A4A5AEF0FF04AEC1FC005FFC7FC49B612FC5F02FCC7B4FCEF3FC00103ED0F E0717E5C717E1307844A1401A2130F17035CA2131F4D5A5C4D5A133F4D5A4A4A5A4D5A01 7F4BC7FC4C5A91C7EA07FC49EC3FF0B812C094C8FC16F83B397DB83F>I<9339FF8001C0 030F13E0037F9038F80380913A01FF807E07913A07F8000F0FDA1FE0EB079FDA3F809038 03BF0002FFC76CB4FCD901FC80495A4948157E495A495A4948153E017F163C49C9FC5B12 01484816385B1207485A1830121F4993C7FCA2485AA3127F5BA312FF90CCFCA41703A25F 1706A26C160E170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C 6C6C5C6CB4495A90393FE00FC0010FB5C9FC010313FC9038007FC03A3D7CBA3B>I<0103 B7FC4916E018F8903B0007F80007FE4BEB00FFF03F80020FED1FC0180F4B15E0F007F002 1F1503A24B15F81801143F19FC5DA2147FA292C8FCA25C18035CA2130119F84A1507A213 0319F04A150FA2010717E0181F4A16C0A2010FEE3F80A24AED7F00187E011F16FE4D5A4A 5D4D5A013F4B5A4D5A4A4A5A057FC7FC017F15FEEE03FC91C7EA0FF049EC7FC0B8C8FC16 FC16C03E397DB845>I<0103B812F05BA290260007F8C7123F4B1407F003E0020F150118 005DA2141FA25D19C0143FA24B1330A2027F1470190092C7126017E05C16014A495A160F 49B6FCA25F9138FC000F01031407A24A6DC8FCA201075C18034A130660010F160693C7FC 4A150E180C011F161C18184A1538A2013F5E18F04A4A5AA2017F15074D5A91C8123F4991 3803FF80B9FCA295C7FC3C397DB83D>I<0103B812E05BA290260007F8C7123F4B140FF0 03C0140F18015DA2141FA25D1980143FA25D1760027F14E095C7FC92C75AA24A1301A24A 495A16070101141F91B6FC94C8FCA2903903FC001F824A130EA21307A24A130CA2010F14 1CA24A90C9FCA2131FA25CA2133FA25CA2137FA291CBFC497EB612C0A33B397DB835>I< DCFF8013E0030F13F0037F9038FC01C0913A01FF803E03913A07FC000F07DA0FE0EB038F DA3FC0903801DF804AC812FFEB01FED903F8157F4948ED3F00495A495A494881017F161E 49C9FC5B12014848161C5B1207485A1818121F4993C7FCA2485AA3127F5BA312FF90CCFC 93387FFFFE93B5FCA29338007FC0715A177F95C7FCA27E5F5F7F123F16016C7E5F6C6C14 036D14071207D803FCEC1EF86C6C143C26007F80EBF07890393FF007E0010FB5EA803001 0349C9FC9038003FE03B3D7DBA41>I<0103B5D8F803B512F8495DA290260007F8C73807 F8004B5DA2020F150F615DA2021F151F615DA2023F153F615DA2027F157F96C7FC92C8FC A24A5D605CA249B7FC60A202FCC7120101031503605CA201071507605CA2010F150F605C A2011F151F605CA2013F153F605CA2017F157F95C8FC91C8FC496C4A7EB690B6FCA34539 7DB845>I<0107B512FCA216F890390007F8005DA2140FA25DA2141FA25DA2143FA25DA2 147FA292C7FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2 133FA25CA2137FA291C8FC497EB6FCA326397DB824>I<0203B512FCA3DA000113006F5A A215015EA315035EA315075EA3150F5EA3151F5EA3153F5EA3157F93C7FCA35D5DA31401 A25DA21403120FD83F805B127FEBC007D8FF805BA24A5AEB001F00FC5C00E0495A006049 C8FC007013FE383801F8381E07F03807FFC0D801FEC9FC2E3B7AB82E>I<0103B500F890 3807FFFC5BA290260007F8C813804BEDFC0019F0020F4B5AF003804B4AC7FC180E021F15 38604B5CEF0380023F4AC8FC170E4B133C1770027F5C4C5ADB0007C9FC160E4A5B167E4A 13FE4B7E01015B92380E7F80ECFC1CED383F010301E07FECFDC04A486C7EECFF00D907FC 6D7E5C4A130783130F707E5C1601011F81A24A6D7EA2013F6F7EA24A143F84137F717E91 C8123F496C81B60107B512C0A26146397DB847>I<0103B6FC5B5E90260007FCC8FC5D5D 140FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA2 130718404A15C0A2010F150118804A1403A2011F16005F4A1406170E013F151E171C4A14 3C177C017F5D160391C7120F49EC7FF0B8FCA25F32397DB839>I<902603FFF893383FFF 80496081D900079438FF80000206DC01BFC7FCA2020E4C5A1A7E020C1606190CDA1C7E16 FE4F5A02181630A20238166162023016C1F00181DA703F158395380303F002601506A202 E0ED0C076202C01518183001016D6C140F06605B028015C0A20103923801801FDD03005B 140092380FC00649173F4D91C8FC01065DA2010E4B5B4D137E130C6F6C5A011C17FEDCE1 805B011802E3C7FCA2013802E6130104EC5C1330ED03F8017016034C5C01F05CD807FC4C 7EB500E0D9C007B512F01680150151397CB851>I<902603FFF891381FFFF8496D5CA2D9 0007030113006FEC007C02061678DA0EFF157081020C6D1460A2DA1C3F15E0705CEC181F 82023815016F6C5C1430150702706D1303030392C7FC02607FA2DAE0015C701306ECC000 8201016E130EEF800C5C163F0103EDC01C041F131891C713E0160F49EDF0381830010614 0717F8010E02031370EFFC60130CEE01FE011C16E004005B011815FF177F133860013015 3FA20170151F95C8FC01F081EA07FCB512E01706A245397DB843>I<4BB4FC031F13F092 38FE01FC913903F0007EDA07C0EB1F80DA1F80EB0FC0023EC7EA07E002FCEC03F0495A49 48EC01F8495A4948EC00FC495A49C912FE49167E13FE49167F1201485AA2485AA2120F5B 001F17FFA2485AA34848ED01FEA400FFEE03FC90C9FCA2EF07F8A2EF0FF0A218E0171F18 C0EF3F806C167F180017FE4C5A6C6C5D1603001F4B5A6D4A5A000FED1F806C6C4AC7FC6D 147E0003EC01F8D801FC495AD8007EEB0FC090263F807FC8FC903807FFF801001380383D 7CBA3F>I<0103B7FC4916E018F8903B0007F80007FC4BEB00FE187F020FED3F80F01FC0 5DA2021F16E0A25DA2143FF03FC05DA2027FED7F80A292C8130018FE4A4A5A604AEC07F0 4D5A0101ED3FC04CB4C7FC91B612FC17E0D903FCCAFCA25CA21307A25CA2130FA25CA213 1FA25CA2133FA25CA2137FA291CBFC497EB6FCA33B397DB835>I<4BB4FC031F13F09238 FE01FC913903F0007EDA07C0EB1F80DA1F80EB0FC0023EC7EA07E002FCEC03F0495A4948 EC01F8495A4948EC00FC495A013F16FE49C9FC13FE187F485A12035B12075B120F4916FF 121FA2485AA34848ED01FEA448C9EA03FCA3EF07F8A218F0170F18E0171F18C0EF3F807E EF7F0017FEDA07C05B6C90391FF001F8903980383803001F496C485A9139E00C0FE0260F C0C0EB1F80D807E1D90E3FC7FC0280137ED803F1EB07F8D801F95C3A007FC00FC0903A3F E07F0003903807FFFE0100018F5BDA000F1306170E171E705A177CEEC1F816FF5FA25F5F 6F5B6F48C7FCED00F8384B7CBA42>I<0103B612F849EDFF8018E0903B0007F8001FF84B EB03FCEF00FE020F157FA24BEC3F80A2021F16C0A25DA2143FF07F805DA2027FEDFF0060 92C7485A4D5A4A4A5A4D5A4AEC1F80057FC7FC0101EC07F891B612E094C8FC9139FC000F C00103EC03F0707E4A6D7E831307177E5C177F010F5D5F5CA2011F1401A25CA2133F1603 4A4A1360A2017F17E019C091C71401496C01011480B61503933900FE0700EF7E0ECAEA1F FCEF07F03B3B7DB83F>I<92391FE00380DBFFFC130002036D5A91390FE01F8F91393F00 07DF027EEB01FE02F81300495A4948147E177C4948143C495AA2011F153891C8FCA34915 30A28094C7FC80806D7E14FEECFFE06D13FE6DEBFFC06D14F06D806D80021F7F02037FEC 003F03037F1500167F163F161FA3120C160FA2001C151F94C7FCA3003C153EA25E003E5D 127E007F4A5A6D495A6DEB0FC0D8F9F0495AD8F0FE01FEC8FC39E03FFFF8010F13E0D8C0 0190C9FC313D7CBA33>I<0003B812FEA25A903AF8003FC00101C0913880007E4848163C 90C7007F141C121E001C92C7FCA2485CA200305C007017180060130112E0485CA21403C7 16005DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA213 01A25CA21303A25CEB0FFC003FB6FC5AA237397EB831>I<003FB56C48B51280485DA226 007F80C7381FF00091C8EA07C0604993C7FCA2491506A20001160E170C5BA20003161C17 185BA20007163817305BA2000F167017605BA2001F16E05F5BA2003F15015F5BA2007F15 0394C8FC90C8FCA25E4815065A160E160C161C161816385E127E5E4B5A6C4A5A4BC9FC6C 6C131E6C6C5B6C6C13F83903F807E06CB55A6C6C48CAFCEB0FF0393B7BB839>I<267FFF FC91383FFFC0B55DA2000390C83807FC006C48ED03E06060000094C7FC5F17065FA25F6D 5DA26D5D17E05F4C5AA24CC8FC6E1306A2013F5C161C16185EA25E6E5BA2011F495A1503 93C9FC1506A25D6E5AA2010F5B157015605DA2ECE18002E3CAFC14F3EB07F614FE5C5CA2 5C5CA26D5AA25C91CBFC3A3B7CB830>I<277FFFFC01B500F890B51280B5FC60000390C7 D807FCC7380FF80001FC4BEC03E000016204035E98C7FC621A0604075DA2040F5DA2041B 5D6216336D02735D1663000003C34A5A83DB01834AC8FC04815CDB0301140603075D1506 030C5DA203185D1970033015606115606D01E04A5A15C090267F01804AC9FC17FEDA0300 14060400130E0206150C020E5D140C4A5DA24A5D18E04A5D715A5C4A92CAFCA26DC85AA2 013E157C1778133C1770133801301560513B7CB84E>I<49B500F890387FFFF095B5FC1A E0D90003018090380FFC004BC713E00201ED07804EC7FC6E6C140E606F5C705B606F6C48 5A4D5A031F91C8FCEEE0065F6F6C5A5F03075B705A16F96FB45A94C9FC6F5AA36F7EA34B 7FED037F9238063FC0150E4B6C7E1538ED700F03E07F15C04A486C7EEC0300020613034A 805C4A6D7E14704A1300494880495A49C86C7E130E011E153F017E4B7ED803FF4B7E007F 01E0011FEBFFC0B5FC6144397EB845>II<91B7 12FCA25B9239E00007F84AC7EA0FF0D903F8EC1FE04AEC3FC04AEC7F804A150049485C91 C7485A4C5A010E4A5A4C5A010C4A5A011C4A5A01185D167F4CC7FC90C7485A4B5A4B5A4B 5A5E151F4B5A4B5A4BC8FC4A5A4A5A4A5A5D140F4A5A4A5A4A48130C4AC7FC495A4A141C 01031518495A494814384948143049481470495A49C812F0495D00011501484814034848 4A5A4848140F4848141F4848EC7F804848EB07FF90B7FCB8FC94C7FC36397BB839>I<14 7E903803FF8090390FC1C38090391F00EFC0017E137F49133F485A4848EB1F8012075B00 0F143F48481400A2485A5D007F147E90C7FCA215FE485C5AA214015D48150CA21403EDF0 1C16181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0F9C03A03 FF007F80D800FCEB1F0026267DA42C>97 D<133FEA1FFFA3C67E137EA313FE5BA312015B A312035BA31207EBE0FCEBE3FF9038E707C0390FFE03E09038F801F001F013F8EBE00048 5A15FC5BA2123F90C7FCA214015A127EA2140312FE4814F8A2140715F05AEC0FE0A215C0 EC1F80143F00781400007C137E5C383C01F86C485A380F07C06CB4C7FCEA01FC1E3B7CB9 24>II<163FED1FFFA3ED007F167EA216FEA216FCA21501A216F8A21503A216F0A21507 A2027E13E0903803FF8790380FC1CF90381F00EF017EEB7FC049133F485A4848131F0007 15805B000F143F485A1600485A5D127F90C7127EA215FE5A485CA21401A248ECF80CA214 03161CEDF0181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0F9 C03A03FF007F80D800FCEB1F00283B7DB92B>II<16F8ED03FEED0F8792381F0F 80ED3E3F167F157CA215FC1700161C4A48C7FCA414035DA414075DA20107B512F0A39026 000FE0C7FC5DA4141F5DA4143F92C8FCA45C147EA514FE5CA413015CA4495AA45C1307A2 5C121E123F387F8F80A200FF90C9FC131E12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA29 >III<14E0EB03F8A21307A314F0EB01C090C7FCAB 13F8EA03FEEA070F000E1380121C121812381230EA701F1260133F00E0130012C05BEA00 7EA213FE5B1201A25B12035BA20007131813E01438000F133013C01470EB806014E014C0 1381EB838038078700EA03FEEA00F815397EB71D>I<150FED3F80A2157FA31600151C92 C7FCABEC0F80EC3FE0ECF0F0903801C0F849487E14005B130E130C131CEB180113380130 5BA2EB0003A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25CA2147EA214 FEA25CA21301001E5B123F387F83F0A238FF87E0495A00FE5BD87C1FC8FCEA707EEA3FF8 EA0FC0214981B722>II< EB0FC0EA03FF5AA2EA001F1480A2133FA21400A25BA2137EA213FEA25BA21201A25BA212 03A25BA21207A25BA2120FA25BA2121FA25BA2123FA290C7FCA25AA2EA7E03A2EAFE0713 0612FCA2130E130C131C1318EA7C38EA3C70EA1FE0EA0780123B7DB919>III<90390F8003F090391FE00FFC903939F03C1F903A70F8700F80 903AE0FDE007C09038C0FF80030013E00001491303018015F05CEA038113015CA2D80003 1407A25CA20107140FA24A14E0A2010F141F17C05CEE3F80131FEE7F004A137E16FE013F 5C6E485A4B5A6E485A90397F700F80DA383FC7FC90387E1FFCEC07E001FEC9FCA25BA212 01A25BA21203A25B1207B512C0A32C3583A42A>112 D<02FC13C0903803FF0190380F83 8390383F01C790397E00EF8049137F485A4848133F000715005B485A001F5C157E485AA2 007F14FE90C75AA3481301485CA31403485CA314075D140F127C141F007E495A003E137F 381F01EF380F839F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5C130190387F FFF0A322357DA425>I<3903E001F83907F807FE390E3C1E07391C3E381F3A183F703F80 0038EBE07F0030EBC0FF00705B00601500EC007E153CD8E07F90C7FCEAC07EA2120013FE 5BA312015BA312035BA312075BA3120F5BA3121F5B0007C9FC21267EA425>I<14FF0103 13C090380F80F090383E00380178131C153C4913FC0001130113E0A33903F000F06D1300 7F3801FFE014FC14FF6C14806D13C0011F13E013039038003FF014071403001E1301127F A24814E0A348EB03C012F800E0EB07800070EB0F006C133E001E13F83807FFE0000190C7 FC1E267CA427>II<13F8 D803FE1438D8070F147C000E6D13FC121C1218003814011230D8701F5C12601503EAE03F 00C001005B5BD8007E1307A201FE5C5B150F1201495CA2151F120349EC80C0A2153F1681 EE0180A2ED7F0303FF130012014A5B3A00F8079F0E90397C0E0F1C90393FFC07F8903907 F001F02A267EA430>I<01F8EB03C0D803FEEB07E0D8070F130F000E018013F0121C1218 0038140700301403D8701F130112601500D8E03F14E000C090C7FC5BEA007E16C013FE5B 1501000115805B150316001203495B1506150E150C151C151815385D00015C6D485A6C6C 485AD97E0FC7FCEB1FFEEB07F024267EA428>I<01F816F0D803FE9138E001F8D8070F90 3801F003000ED9800314FC121C12180038020713010030EDE000D8701F167C1260030F14 3CD8E03F163800C001005B5BD8007E131F183001FE5C5B033F1470000117604991C7FCA2 18E000034A14C049137E17011880170318005F03FE1306170E000101015C01F801BF5B3B 00FC039F8070903A7E0F0FC0E0903A1FFC03FFC0902703F0007FC7FC36267EA43B>I<90 3907E001F090391FF807FC9039783E0E0F9039E01F1C1FD801C09038383F803A03800FF0 7F0100EBE0FF5A000E4A1300000C157E021F133C001C4AC7FC1218A2C7123FA292C8FCA2 5CA2147EA214FEA24A130CA20101141C001E1518003F5BD87F81143801835C00FF156001 0714E03AFE0E7C01C0D87C1C495A2778383E0FC7FC391FF00FFC3907C003F029267EA42F >I<13F8D803FE1470D8070F14F8000EEB8001121C121800381403003015F0EA701F1260 013F130700E0010013E012C05BD8007E130F16C013FE5B151F000115805BA2153F000315 005BA25D157EA315FE5D1401000113033800F80790387C1FF8EB3FF9EB0FE1EB00035DA2 000E1307D83F805B007F495AA24A5A92C7FCEB003E007C5B00705B6C485A381E07C06CB4 C8FCEA01FC25367EA429>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA cmr10 10 97 /FA 97 128 df<1506150FA24B7EA24B7EA24B7EA2EDDFF0A29138018FF8A291380307FC A291380603FEA291380E01FF140CDA1C007F141802386D7E143002706D7E146002E06D7E 5C01016E7E5C01036E7E91C7FC496E7E1306010E6E7E130C011C6E7F131801386F7E1330 01706F7E136001E06F7E5B170F484882170748C97F17030006831701488383481880001F B9FC4818C0A24818E0A2BA12F0A23C3C7CBB45>1 D<003FB712FCA60030C9120C007016 0EA200601606A4CBFCA701C01403A490B7FCA601C0C71203A490CAFCA900C01603A56C16 07A200601606007FB712FEA630397DB837>4 DIII<011FB512FEA39026001FFEC8FCEC07F8A8EC3FFE 0103B512E0D91FF713FC90397F07F87F01FCEC1F80D803F8EC0FE0D807F06E7ED80FE06E 7E001F82D83FC06E7EA2007F8201808000FF1780A7007F170001C05C003F5EA2D81FE04A 5A000F5ED807F04A5AD803F84A5AD800FCEC1F80017F027FC7FC90391FF7FFFC0103B512 E09026003FFEC8FCEC07F8A8EC1FFE011FB512FEA331397BB83C>I10 DI III<133C137EA213FE1201EA03FC13F0EA07E0EA0FC0EA1F80EA1E005A5A5A12C00F0F 6FB92A>19 D22 D<001C131C007F137F39FF80FF80A26D13C0A3 007F137F001C131C00001300A40001130101801380A20003130301001300485B00061306 000E130E485B485B485B006013601A197DB92A>34 D<141FEC7FC0903801F0E0903803C0 600107137090380F803090381F00381518A25BA2133E133F15381530A215705D5D140190 381F838092CAFC1487148E02DC49B51280EB0FF85C4A9039003FF8000107ED0FC06E5D71 C7FC6E140E010F150CD91DFC141C01391518D970FE143801E015302601C07F1470D80380 5D00076D6C5BD80F00EBC00148011F5C4890380FE003003E6E48C8FC007E903807F80602 03130E00FE6E5A6E6C5A1400ED7F706C4B13036F5A6F7E6C6C6D6C5B7013066C6C496C13 0E6DD979FE5B281FF001F07F133C3C07F80FE03FC0F86CB539800FFFF0C69026FE000313 C0D91FF0D9007FC7FC393E7DBB41>38 D<121C127FEAFF80A213C0A3127F121C1200A412 011380A2120313005A1206120E5A5A5A12600A1979B917>I<146014E0EB01C0EB0380EB 0700130E131E5B5BA25B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2 127CA67EA3121EA2121F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB 01C0EB00E01460135278BD20>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378 A2137C133C133E131EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A2 5B131EA2133E133C137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD 20>I<15301578B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41> 43 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A 5A5A12600A19798817>II<121C127FEAFF80A5EA7F00121C0909 798817>I<150C151E153EA2153C157CA2157815F8A215F01401A215E01403A215C01407 A21580140FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA2 5C130FA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A2 5B120FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD2A>IIIII<1538A2157815F8A214 0114031407A2140F141F141B14331473146314C313011483EB030313071306130C131C13 1813301370136013C01201EA038013005A120E120C5A123812305A12E0B712F8A3C73803 F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C9038F003F890B5FC5D 5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F803907E007E090388003 F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300485C12E000605C12 700030495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7FC38007FFCEB1FE0 213A7CB72A>II<12301238123E003FB612E0A316C0 5A168016000070C712060060140E5D151800E01438485C5D5DC712014A5A92C7FC5C140E 140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133FA5137FA96DC8FC13 1E233B7BB82A>III<121C127FEA FF80A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C092479A317>I<121C127FEA FF80A5EA7F00121CC7FCB2121C127F5A1380A4127F121D1201A412031300A25A1206A212 0E5A121812385A1260093479A317>I<007FB812F8B912FCA26C17F8CCFCAE007FB812F8 B912FCA26C17F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC06 3FA2020E7FEC0C1FA2021C7FEC180FA202387FEC3007A202707FEC6003A202C07F1501A2 D901807F81A249C77F167FA20106810107B6FCA24981010CC7121FA2496E7EA3496E7EA3 496E7EA213E0707E1201486C81D80FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF807E07903A03 FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F12014848151F48 48150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F6DED0180A312 3F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F15386D6C5CD91FE0 5C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F002011380313D 7BBA3C>III< B812F8A30001903880001F6C90C71201EE00FC177C173C171CA2170CA4170E1706A2ED01 80A21700A41503A21507151F91B5FCA3EC001F15071503A21501A692C8FCAD4813C0B612 C0A32F397DB836>III I<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F80EAFFC0A44A5A1380D87F005B 0070131F6C5C6C495A6C49C7FC380781FC3801FFF038007F80233B7DB82B>III< B5933807FFF86E5DA20001F0FC002600DFC0ED1BF8A2D9CFE01533A3D9C7F01563A3D9C3 F815C3A2D9C1FCEC0183A3D9C0FEEC0303A2027F1406A36E6C130CA36E6C1318A26E6C13 30A36E6C1360A26E6C13C0A3913901FC0180A3913900FE0300A2ED7F06A3ED3F8CA2ED1F D8A3ED0FF0A3486C6D5A487ED80FFC6D48497EB500C00203B512F8A2ED018045397DB84C >I IIIIII<003FB812E0A3D9C003EB001F273E0001FE130348EE 01F00078160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011F B612E0A335397DB83C>IIII89 D<003FB7FCA39039FC0001FE01C0130349495A003EC7FC003C4A5A5E0038141F00 784A5A12704B5A5E006014FF4A90C7FCA24A5A5DC712074A5AA24A5A5D143F4A5AA24A5A 92C8FC5B495AA2495A5C130F4948EB0180A2495A5C137F495A16034890C7FC5B1203485A EE0700485A495C001F5D48485C5E4848495A49130FB8FCA329397BB833>II<3901800180000313033907000700000E130E 485B0018131800381338003013300070137000601360A200E013E0485BA400CE13CE39FF 80FF806D13C0A3007F137FA2393F803F80390E000E001A1974B92A>II<13101338137C13FE487E3803C780380783C0380F 01E0381E00F04813780070131C48130E00401304170D77B92A>I<121E123FEA7F80A2EA FFC0EA7F80A2EA3F00121E0A097AB717>IIIIIII<147E903803FF8090380FC1E0EB1F8790383F0FF0137E A213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19 >IIIIIII<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F 83803F3D0FF3800FC7001F802603F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3 A3486C496CEB1FE0B500C1B50083B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF0 3F9039F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C497EB500C1B5 1280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3 801F803A07F7000FC0D803FEEB07E049EB03F04914F849130116FC150016FEA3167FAA16 FEA3ED01FCA26DEB03F816F06D13076DEB0FE001F614C09039F7803F009038F1E07E9038 F0FFF8EC1FC091C8FCAB487EB512C0A328357EA42E>II<3807E01F00FFEB7FC09038 E1E3E09038E387F0380FE707EA03E613EE9038EC03E09038FC0080491300A45BB3A2487E B512F0A31C257EA421>II<1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FC B215C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220>IIIIII<00 3FB512FCA2EB8003D83E0013F8003CEB07F00038EB0FE012300070EB1FC0EC3F80006013 7F150014FE495AA2C6485A495AA2495A495A495AA290387F000613FEA2485A485A000714 0E5B4848130C4848131CA24848133C48C7127C48EB03FC90B5FCA21F247EA325>III126 D<001C131C007F137F39FF80FF80A5397F007F00001C131C190978B72A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FB cmbx10 10 57 /FB 57 123 df<141C143C14F8EB01F0EB03E01307EB0FC0EB1F8014005B137E13FE5B12 015B1203A2485AA2120F5B121FA25B123FA4485AA512FFB1127FA56C7EA4121F7FA2120F 7F1207A26C7EA212017F12007F137E7F7F1480EB0FC0EB07E01303EB01F0EB00F8143C14 1C165377BD25>40 D<12E07E127C7E7E7F6C7E6C7E12037F6C7E7F12007F137E137FA2EB 3F80A214C0131F14E0A2130F14F0A4EB07F8A514FCB114F8A5EB0FF0A414E0131FA214C0 133F1480A2EB7F00A2137E13FE5B12015B485A5B1207485A485A90C7FC123E5A12F05A16 537BBD25>I45 DI<49B4FC010F13E0017F13FC9038FF83FE4848C67E4848EB 7F804848EB3FC04848EB1FE0A2001F15F0A24848EB0FF8A3007F15FCA500FF15FEB3007F 15FCA4003F15F8A26D131F001F15F0A2000F15E06D133F000715C06C6CEB7F806C6CEBFF 003900FF83FE6DB45A011F13F0010190C7FC27387CB630>48 D<141E143E14FE1307133F B5FCA313CFEA000FB3B3A6007FB61280A4213779B630>IIII<001C15C0D81F80130701F8137F90B61280A21600 5D5D15F05D15804AC7FC14F090C9FCA8EB07FE90383FFFE090B512F89038FC07FC9038E0 03FFD98001138090C713C0120EC813E0157F16F0A216F8A21206EA3F80EA7FE012FF7FA4 4914F0A26C4813FF90C713E0007C15C06C5B6C491380D9C0071300390FF01FFE6CB512F8 000114E06C6C1380D90FF8C7FC25387BB630>II<123C123EEA3FE090B71280A41700485D5E5E5EA25E007CC7EA0FC000784A5A 4BC7FC00F8147E48147C15FC4A5A4A5AC7485A5D140F4A5A143F92C8FC5C147E14FE1301 A2495AA31307A2130F5CA2131FA5133FA96D5A6D5A6D5A293A7BB830>I<49B47E010F13 F0013F13FC9038FE01FF3A01F8007F804848EB3FC04848EB1FE0150F485AED07F0121FA2 7FA27F7F01FEEB0FE0EBFF809138E01FC06CEBF03F02FC13809138FF7F006C14FC6C5C7E 6C14FE6D7F6D14C04914E048B612F0EA07F848486C13F8261FE01F13FC383FC007EB8001 007F6D13FE90C7123F48140F48140715031501A21500A216FC7E6C14016D14F86C6CEB03 F06D13076C6CEB0FE0D80FFEEB7FC00003B61200C614FC013F13F00103138027387CB630 >III65 DIIIIIIII75 DIII80 D83 D<003FB91280A4D9F800EBF003D87FC09238007FC049161F007EC715 0FA2007C1707A200781703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B387DB742 >I86 D97 D<13FFB5FCA412077EAF4AB47E020F13F0023F13FC9138FE03FFDAF000 13804AEB7FC00280EB3FE091C713F0EE1FF8A217FC160FA217FEAA17FCA3EE1FF8A217F0 6E133F6EEB7FE06E14C0903AFDF001FF80903AF8FC07FE009039F03FFFF8D9E00F13E0D9 C00390C7FC2F3A7EB935>I<903801FFC0010F13FC017F13FFD9FF8013802603FE0013C0 48485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F123FED01 E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F00101138023 257DA42A>I I<903803FF80011F13F0017F13FC3901FF83FE3A03FE007F804848133F484814C0001FEC 1FE05B003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C7E1678121F 6C6C14F86D14F000071403D801FFEB0FE06C9038C07FC06DB51200010F13FC010113E025 257DA42C>II<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F260FF801131F48486C13 8F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF001F5D6C6C4890C7FC3907 FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E90B512F8EDFF8016E0 6C15F86C816C815A001F81393FC0000F48C8138048157F5A163FA36C157F6C16006D5C6C 6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC010713F02B377DA530>I< 13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03FFDA3C01138014784A7E4A 14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>II<13FFB5 FCA412077EAF92380FFFE0A4923803FC0016F0ED0FE0ED1F804BC7FC157E5DEC03F8EC07 E04A5A141FEC7FE04A7E8181A2ECCFFEEC0FFF496C7F806E7F6E7F82157F6F7E6F7E8215 0F82B5D8F83F13F8A42D3A7EB932>107 D<13FFB5FCA412077EB3B3ACB512FCA4163A7D B91B>I<01FED97FE0EB0FFC00FF902601FFFC90383FFF80020701FF90B512E0DA1F8190 3983F03FF0DA3C00903887801F000749DACF007F00034914DE6D48D97FFC6D7E4A5CA24A 5CA291C75BB3A3B5D8FC1FB50083B512F0A44C257DA451>I<01FEEB7FC000FF903803FF F8020F13FE91381F03FFDA3C011380000713780003497E6D4814C05CA25CA291C7FCB3A3 B5D8FC3F13FFA430257DA435>I<903801FFC0010F13F8017F13FFD9FF807F3A03FE003F E048486D7E48486D7E48486D7EA2003F81491303007F81A300FF1680A9007F1600A3003F 5D6D1307001F5DA26C6C495A6C6C495A6C6C495A6C6C6CB45A6C6CB5C7FC011F13FC0101 13C029257DA430>I<9039FF01FF80B5000F13F0023F13FC9138FE07FFDAF00113800003 496C13C00280EB7FE091C713F0EE3FF8A2EE1FFCA3EE0FFEAA17FC161FA217F8163F17F0 6E137F6E14E06EEBFFC0DAF00313809139FC07FE0091383FFFF8020F13E0020390C7FC91 C9FCACB512FCA42F357EA435>I<49B4EB0780010FEBE00F013FEBF81F9039FFC07C3F00 03EB803E3A07FE000F7F4848EB07FF121F497F123F497F127FA25B12FFAA6C7EA36C7E5D 6C7E000F5C6C6C5B6C6C133F6CEBC0FD39007FFFF1011F13C10101130190C7FCAC037F13 FEA42F357DA432>I<9038FE03F000FFEB0FFEEC3FFF91387C7F809138F8FFC000075B6C 6C5A5CA29138807F80ED3F00150C92C7FC91C8FCB3A2B512FEA422257EA427>I<90383F F0383903FFFEF8000F13FF381FC00F383F0003007E1301007C130012FC15787E7E6D1300 13FCEBFFE06C13FCECFF806C14C06C14F06C14F81203C614FC131F9038007FFE140700F0 130114007E157E7E157C6C14FC6C14F8EB80019038F007F090B512C000F8140038E01FF8 1F257DA426>I<130FA55BA45BA25B5BA25A1207001FEBFFE0B6FCA3000390C7FCB21578 A815F86CEB80F014816CEBC3E090383FFFC06D1380903803FE001D357EB425>I<01FFEC 3FC0B5EB3FFFA4000714016C80B3A35DA25DA26C5C6E4813E06CD9C03E13FF90387FFFFC 011F13F00103138030257DA435>IIIII<003FB612C0A3D9F0031380EB800749481300003E5C003C495A007C133F5D00 78495A14FF5D495B5BC6485B92C7FC495A131F5C495A017FEB03C0EBFFF014E04813C05A EC80074813005A49EB0F80485A003F141F4848133F9038F001FFB7FCA322257DA42A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmbx12 14.4 54 /FC 54 123 df45 DI<913803FFC0023F13FC91B6FC010315C0010F0181 13F0903A1FFC003FF849486D7E49486D7E49486D7E48496D138048496D13C0A24817E048 90C813F0A34817F8A24817FC49157FA3007F17FEA600FF17FFB3A5007F17FEA6003F17FC A26D15FFA26C17F8A36C17F0A26C6D4913E0A26C6D4913C06C17806E5B6C6D4913006D6C 495AD91FFCEB3FF8903A0FFF81FFF06D90B55A01011580D9003F01FCC7FC020313C0384F 7BCD43>48 D<157815FC14031407141F14FF130F0007B5FCB6FCA2147F13F0EAF800C7FC B3B3B3A6007FB712FEA52F4E76CD43>II<91380FFF C091B512FC0107ECFF80011F15E090263FF8077F9026FF800113FC4848C76C7ED803F86E 7E491680D807FC8048B416C080486D15E0A4805CA36C17C06C5B6C90C75AD801FC1680C9 FC4C13005FA24C5A4B5B4B5B4B13C04B5BDBFFFEC7FC91B512F816E016FCEEFF80DA0007 13E0030113F89238007FFE707E7013807013C018E07013F0A218F8A27013FCA218FEA2EA 03E0EA0FF8487E487E487EB57EA318FCA25E18F891C7FC6C17F0495C6C4816E001F04A13 C06C484A1380D80FF84A13006CB44A5A6CD9F0075BC690B612F06D5D011F1580010302FC C7FCD9001F1380374F7ACD43>I<177C17FEA2160116031607160FA2161F163F167FA216 FF5D5DA25D5DED1FBFED3F3F153E157C15FCEC01F815F0EC03E01407EC0FC01580EC1F00 5C147E147C5C1301495A495A5C495A131F49C7FC133E5B13FC485A5B485A1207485A485A 90C8FC123E127E5ABA12C0A5C96C48C7FCAF020FB712C0A53A4F7CCE43>III<121F7F7FEBFF8091B81280A45A1900606060A2606060485F01 80C86CC7FC007EC95A4C5A007C4B5A5F4C5A160F4C5A484B5A4C5A94C8FC16FEC812014B 5A5E4B5A150F4B5AA24B5AA24B5A15FFA24A90C9FCA25C5D1407A2140FA25D141FA2143F A4147F5DA314FFA55BAC6D5BA2EC3FC06E5A395279D043>I<913807FFC0027F13FC0103 B67E010F15E090261FFC0113F8903A3FE0003FFCD97F80EB0FFE49C76C7E48488048486E 1380000717C04980120F18E0177FA2121F7FA27F7F6E14FF02E015C014F802FE4913806C 7FDBC00313009238F007FE6C02F85B9238FE1FF86C9138FFBFF06CEDFFE017806C4BC7FC 6D806D81010F15E06D81010115FC010781011F81491680EBFFE748018115C048D9007F14 E04848011F14F048487F48481303030014F8484880161F4848020713FC1601824848157F 173FA2171FA2170FA218F8A27F007F17F06D151FA26C6CED3FE0001F17C06D157F6C6CED FF806C6C6C010313006C01E0EB0FFE6C01FCEBFFFC6C6CB612F06D5D010F1580010102FC C7FCD9000F13C0364F7ACD43>I<91380FFF8091B512F8010314FE010F6E7E4901037F90 267FF8007F4948EB3FF048496D7E484980486F7E484980824817805A91C714C05A7013E0 A218F0B5FCA318F8A618FCA46C5DA37EA25E6C7F6C5DA26C5D6C7F6C6D137B6C6D13F390 387FF803011FB512E36D14C30103028313F89039007FFE03EC00401500A218F05EA3D801 F816E0487E486C16C0487E486D491380A218005E5F4C5A91C7FC6C484A5A494A5A49495B 6C48495BD803FC010F5B9027FF807FFEC7FC6C90B55A6C6C14F06D14C0010F49C8FC0100 13F0364F7ACD43>II<17 1F4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C7FA24C7FA34C8083047F80167E8304FE80 4C7E03018116F8830303814C7E03078116E083030F814C7E031F81168083033F8293C77E 4B82157E8403FE824B800201835D840203834B800207835D844AB87EA24A83A3DA3F80C8 8092C97E4A84A2027E8202FE844A82010185A24A820103854A82010785A24A82010F855C 011F717FEBFFFCB600F8020FB712E0A55B547BD366>65 DI<932601FFFCEC01C0047FD9FFC013030307B600F8130703 3F03FE131F92B8EA803F0203DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01 800203B5FC494848C9FC4901F8824949824949824949824949824990CA7E494883A24849 83485B1B7F485B481A3FA24849181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F3 07C07EA36C7FA21B0F6C6D1980A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A 6D6D16036D6D4C5A6D6D4C5A6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE02 0F01FEEC1FF80203903AFFE001FFF0020091B612C0033F93C8FC030715FCDB007F14E004 0101FCC9FC525479D261>II II<932601FFFCEC01C0047FD9FFC01303 0307B600F81307033F03FE131F92B8EA803F0203DAE003EBC07F020F01FCC7383FF0FF02 3F01E0EC0FF94A01800203B5FC494848C9FC4901F8824949824949824949824949824990 CA7E494883A2484983485B1B7F485B481A3FA24849181FA3485B1B0FA25AA298C8FC5CA2 B5FCAE6C057FB712E0A280A36C94C7003FEBC000A36C7FA36C7FA27E6C7FA26C7F6C7FA2 6D7E6D7F6D7F6D6D5E6D7F6D01FC93B5FC6D13FF6D6C6D5C6E01F0EC07FB020F01FEEC1F F10203903AFFF001FFE0020091B6EAC07F033FEE001F030703FC1307DB007F02E0130104 0149CAFC5B5479D26A>III75 DIII80 D82 D<91260FFF80130791B500F85B010702FF5B011FEDC03F49EDF07F9026FFFC006D5A4801 E0EB0FFD4801800101B5FC4848C87E48488149150F001F824981123F4981007F82A28412 FF84A27FA26D82A27F7F6D93C7FC14C06C13F014FF15F86CECFF8016FC6CEDFFC017F06C 16FC6C16FF6C17C06C836C836D826D82010F821303010082021F16801400030F15C0ED00 7F040714E01600173F050F13F08383A200788200F882A3187FA27EA219E07EA26CEFFFC0 A27F6D4B13806D17006D5D01FC4B5A01FF4B5A02C04A5A02F8EC7FF0903B1FFFC003FFE0 486C90B65AD8FC0393C7FC48C66C14FC48010F14F048D9007F90C8FC3C5479D24B>I<00 3FBC1280A59126C0003F9038C0007F49C71607D87FF8060113C001E08449197F49193F90 C8171FA2007E1A0FA3007C1A07A500FC1BE0481A03A6C994C7FCB3B3AC91B912F0A55351 7BD05E>I97 DI<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A1FFE0001 FFD97FFC491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F1300705A48 92C8FC5BA312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F806C6DEC 3F006C6D147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A01001580023F49C7FC 020113E033387CB63C>I<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91 B6FC010315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980 485B83485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C 6D49B5FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9 003F13F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC010715C04901 817F903A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191 C7FC4817E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01 F06E14037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE 903A0FFFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>IIII<137F497E 000313E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512017EB3B3 A6B612E0A51B547BD325>I 107 DIII<913801FFE0021F13FE91B612C0010315F0010F9038 807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F48496D7F4A147F48834890C8 6C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA2 6C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE011F90C7FC903A0FFF80 7FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I<903A3FF001FFE0B5010F 13FE033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9FFE0EB1FFFC602806D7F 92C76C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A0060A36118FFA2615F61 6E4A5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FBFE075B02F8B612E06F14 80031F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I<912601FFE0EB0780021F 01F8130F91B500FE131F0103ECFF80010F9039F03FC03F499039800FE07F903A7FFE0003 F04948903801F8FF4849EB00FD4849147F4A805A4849805A4A805AA291C87E5AA35B12FF AC6C7EA37EA2806C5EA26C6D5CA26C6D5C6C6D5C6C93B5FC6C6D5B6D6C5B6DB4EB0FEF01 0F9038C07FCF6D90B5120F010114FED9003F13F80203138091C8FCB1040FB61280A5414D 7CB547>I<90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F83FFC00 03D9E3E013FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90C7FCA5 5CB3AAB612FCA52F367CB537>I<903903FFF00F013FEBFE1F90B7FC120348EB003FD80F F81307D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C 13FF15F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714E0EC00 3F030713F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6DEC3F80 01F8EC7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C 387CB635>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6 FCB8FCA426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0 FC6DEBFFF86D6C5B021F5B020313802A4D7ECB34>II< B600F00107B5FCA5000101F8C8EA7FE06C6DED3F00A2017F163E6E157E013F167C6E15FC 6D5E6F13016D5E8117036D5E6F13076D5E6F130F6D5E6F131F6D93C7FC815F6E6C133E17 7E023F147C6F13FC6E5C16816E5C16C3A26EEBE3E016E76E5C16FF6E5CA26E91C8FCA26F 5AA36F5AA26F5AA26F5AA26F5A6F5A40367DB447>I<007FB500F090387FFFFEA5C66C48 C7000F90C7FC6D6CEC07F86D6D5C6D6D495A6D4B5A6F495A6D6D91C8FC6D6D137E6D6D5B 91387FFE014C5A6E6C485A6EEB8FE06EEBCFC06EEBFF806E91C9FCA26E5B6E5B6F7E6F7E A26F7F834B7F4B7F92B5FCDA01FD7F03F87F4A486C7E4A486C7E020F7FDA1FC0804A486C 7F4A486C7F02FE6D7F4A6D7F495A49486D7F01076F7E49486E7E49486E7FEBFFF0B500FE 49B612C0A542357EB447>120 DI<001FB8FC1880A3912680 007F130001FCC7B5FC01F0495B495D49495B495B4B5B48C75C5D4B5B5F003E4A90C7FC92 B5FC4A5B5E4A5B5CC7485B5E4A5B5C4A5B93C8FC91B5FC495B5D4949EB0F805B495B5D49 5B49151F4949140092C7FC495A485E485B5C485E485B4A5C48495B4815074849495A91C7 12FFB8FCA37E31357CB43C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FD cmr12 12 12 /FD 12 118 df<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>46 D66 D75 D<49B41303010FEBE007013F13 F89039FE00FE0FD801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E8112 7E82127C00FC81A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEB FFE06C14FC6C14FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380 816F13C0167F163F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C1600 6C5D6D147ED8FBC05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC 39C0003FF02B487BC536>83 D101 D103 D105 D107 D<3901FC01FE00FF903807FFC091381E07F091 383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CEC FF80B5D8F83F13FEA32F2C7DAB36>110 D<3903F803F000FFEB1FFCEC3C3EEC707F0007 EBE0FF3803F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7D AB26>114 D<90383FE0183901FFFC383907E01F78390F0003F8001E1301481300007C14 78127800F81438A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C 14E0C614F0011F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C14 7C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I< D801FC147F00FFEC3FFFA300071401000380000181B3A85EA35DA212006D5B017E903807 7F80017F010E13C06D011C13FE90380FC078903803FFF09026007F8013002F2D7DAB36> 117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FE cmr17 17.28 16 /FE 16 119 df12 D68 DI 72 D80 D97 D101 D<133C13FF487F487FA66C5B6C90C7FC133C90C8FCB3A2 EB03C0EA07FF127FA41201EA007FA2133FB3B3AC497E497EB612E0A41B5F7DDE23>105 D108 DIII< 9139FFE00180010FEBFC03017FEBFF073A01FF001FCFD803F8EB03EFD807E0EB01FF4848 7F4848147F48C8123F003E151F007E150F127CA200FC1507A316037EA27E7F6C7E6D91C7 FC13F8EA3FFE381FFFF06CEBFF806C14F86C14FF6C15C06C6C14F0011F80010714FED900 7F7F02031480DA003F13C01503030013E0167F00E0ED1FF0160F17F86C15071603A36C15 01A37EA26C16F016037E17E06D14076DEC0FC06D1580D8FDF0141FD8F8F8EC7F00013E14 FC3AF01FC00FF80107B512E0D8E001148027C0003FF8C7FC2D417DBF34>115 D<1438A71478A414F8A31301A31303A21307130F131FA2137F13FF1203000F90B6FCB8FC A3260007F8C8FCB3AE17E0AE6D6CEB01C0A316036D6C148016076D6C14006E6C5A91383F C01E91381FF07C6EB45A020313E09138007F802B597FD733>III E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 userdict/PStoPSxform PStoPSmatrix matrix currentmatrix matrix invertmatrix matrix concatmatrix matrix invertmatrix put %%EndSetup %%Page: (0,1) 1 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 1 0 bop 933 872 a FE(Hamiltonian)46 b(PDEs)d(in)i(\014nite)f(v)l(olume) 1588 1112 y FD(Sergei)32 b(B.)h(Kuksin)515 1469 y FC(Con)l(ten)l(ts)515 1652 y FB(1)76 b(In)m(tro)s(duction)2166 b(2)515 1835 y(2)76 b(Symplectic)31 b(Hilb)s(ert)f(scales)i(and)639 1934 y(Hamiltonian)e(equations)1746 b(2)639 2034 y FA(2.1)84 b(Hilb)r(ert)28 b(scales)f(and)g(their)h(morphisms)48 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.) 131 b(2)639 2134 y(2.2)84 b(Symplectic)28 b(structures)84 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)f(.)h(.)f(.)h(.)131 b(5)639 2233 y(2.3)84 b(Hamiltonian)28 b(equations)45 b(.)c(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)131 b(5)515 2416 y FB(3)76 b(Basic)32 b(theorems)e(on)h(Hamiltonian)f (systems)1017 b(8)515 2599 y(4)76 b(Lax-in)m(tegrable)33 b(equations)1616 b(10)639 2698 y FA(4.1)84 b(General)27 b(discussion)67 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(10)639 2798 y(4.2)84 b(Kortew)n(eg{de)25 b(V)-7 b(ries)28 b(equation)44 b(.)e(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(11)639 2897 y(4.3)84 b(Other)27 b(examples)38 b(.)k(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(12)515 3080 y FB(5)76 b(KAM)32 b(for)g(PDEs)1988 b(13)639 3180 y FA(5.1)84 b(An)28 b(abstract)f(KAM-theorem)74 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)h(.)f(.)h(.)90 b(13)639 3279 y(5.2)84 b(Applications)28 b(to)f(1D)h(HPDEs)72 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(17)639 3379 y(5.3)84 b(Multiple)29 b(sp)r(ectrum)69 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)90 b(19)639 3479 y(5.4)84 b(Space-m)n(ultidimensional)27 b(problems)42 b(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)90 b(19)639 3578 y(5.5)84 b(P)n(erturbations)26 b(of)h(in)n(tegrable)g(equations)33 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(20)639 3678 y(5.6)84 b(Small)28 b(amplitude)g(solutions)f(of)g(HPDEs)48 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(24)515 3861 y FB(6)76 b(Around)32 b(the)g(Nekhoroshev)g(theorem)1209 b(25)515 4043 y(7)76 b(In)m(v)-5 b(arian)m(t)34 b(Gibbs)d(measures)1575 b(27)515 4226 y(8)76 b(The)32 b(non-squeezing)f(phenomenon)639 4325 y(and)i(symplectic)d(capacit)m(y)1649 b(28)639 4425 y FA(8.1)84 b(The)28 b(Gromo)n(v)e(theorem)82 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)f(.)h(.)90 b(28)639 4525 y(8.2)84 b (In\014nite-dimensional)28 b(case)37 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(29)639 4624 y(8.3)84 b(Examples)65 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)f(.)h(.)90 b(31)639 4724 y(8.4)84 b(Symplectic)28 b(capacit)n(y)80 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(32)515 4907 y FB(9)76 b(The)32 b(squeezing)f(phenomenon)f(and)639 5006 y(the)i(essen)m(tial)f(part)h(of)g(the)g(phase-space)1126 b(33)1926 5255 y FA(1)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 2 1 bop 515 523 a FB(10)28 b(App)s(endix:)49 b(F)-8 b(amilies)33 b(of)j(p)s(erio)s(dic)e(orbits)h(in)g(rev)m(ersible)g(PDEs.)53 b(By)639 623 y(Dario)32 b(Bam)m(busi)2017 b(36)639 722 y FA(10.1)42 b(In)n(tro)r(duction)25 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)f(.)h(.)90 b(36)639 822 y(10.2)42 b(An)28 b(abstract)f(theorem)g(for)g(nonresonan)n(t)f(PDEs)54 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(36)639 922 y(10.3)42 b(The)28 b(resonan)n(t)e(case)84 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(39)639 1021 y(10.4)42 b(W)-7 b(eak)n(ening)27 b(the)h(nonresonance)e(condition)36 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(41)639 1121 y(10.5)42 b(The)28 b(w)n(ater)e(w)n(a)n(v)n(e)g(problem) 34 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(42)515 1395 y FC(1)134 b(In)l(tro)t(duction)515 1577 y FA(In)19 b(this)g(w)n(ork)f(w)n(e)h (discuss)f(qualitativ)n(e)g(prop)r(erties)g(of)h(solutions)g(for)f (Hamiltonian)h(partial)515 1677 y(di\013eren)n(tial)j(equations)g(in)h (the)h(\014nite)f(v)n(olume)f(case.)35 b(That)22 b(is,)i(when)f(the)g (space-v)-5 b(ariable)515 1776 y Fz(x)27 b FA(b)r(elongs)e(to)i(a)f (\014nite)h(domain)f(and)g(appropriate)f(b)r(oundary)g(conditions)h (are)g(sp)r(eci\014ed)515 1876 y(on)k(the)i(domain's)e(b)r(oundary)g (\(or)g Fz(x)h FA(b)r(elongs)g(to)f(the)i(whole)e(space,)h(but)g(the)h (equation)515 1976 y(con)n(tains)d(a)g(p)r(oten)n(tial)h(term,)g(where) g(the)g(p)r(oten)n(tial)g(gro)n(wths)e(to)i(in\014nit)n(y)g(as)g Fy(j)p Fz(x)p Fy(j)d(!)g(1)p FA(,)515 2075 y(cf.)35 b(b)r(elo)n(w)21 b(Example)g(5.5)g(in)h(section)f(5.2\).)35 b(Most)21 b(of)h(these)g(prop)r(erties)f(ha)n(v)n(e)f(analogies)g(in)515 2175 y(the)32 b(classical)f(\014nite-dimensional)h(Hamiltonian)g(mec)n (hanics.)50 b(In)33 b(the)f(in\014nite-v)n(olume)515 2275 y(case)22 b(prop)r(erties)h(of)h(the)g(equations)e(b)r(ecome)i (rather)e(di\013eren)n(t)i(due)g(to)f(the)h(phenomenon)515 2374 y(of)j(radiation,)g(and)g(w)n(e)g(do)h(not)f(touc)n(h)h(them)g (here.)639 2474 y(Our)f(bibliograph)n(y)f(is)i(b)n(y)f(no)g(means)g (complete.)639 2673 y FB(Notation.)43 b FA(By)30 b Fx(T)1269 2643 y Fw(n)1344 2673 y FA(w)n(e)f(denote)h(the)g(torus)g Fx(T)2155 2643 y Fw(n)2226 2673 y FA(=)d Fx(R)2372 2643 y Fw(n)2409 2673 y Fz(=)p FA(2)p Fz(\031)s Fx(Z)2604 2643 y Fw(n)2673 2673 y FA(and)j(write)g Fx(T)3108 2643 y Fv(1)3171 2673 y FA(=)d Fz(S)3319 2643 y Fv(1)3356 2673 y FA(;)515 2773 y(b)n(y)d Fx(R)681 2743 y Fw(n)681 2793 y Fv(+)767 2773 y FA({)g(the)h(op)r(en)g(p)r(ositiv)n(e)f(o)r (ctan)n(t)g(in)h Fx(R)1878 2743 y Fw(n)1929 2773 y FA(;)h(b)n(y)e Fx(Z)2151 2785 y Fv(0)2207 2773 y FA({)g(the)h(set)g(of)f(nonzero)g(in) n(tegers.)34 b(By)515 2872 y Fz(B)578 2884 y Fw(\016)614 2872 y FA(\()p Fz(x)p FA(;)14 b Fz(X)7 b FA(\))33 b(w)n(e)f(denote)g (an)g(op)r(en)g Fz(\016)s FA(-ball)g(in)h(a)f(space)f Fz(X)7 b FA(,)33 b(cen)n(tred)f(at)g Fz(x)f Fy(2)h Fz(X)7 b FA(.)50 b(Abusing)515 2972 y(notation,)25 b(w)n(e)g(denote)f(b)n(y)h Fz(x)h FA(b)r(oth)f(the)h(space-v)-5 b(ariable)23 b(and)h(an)h(elemen)n (t)g(of)g(an)g(abstract)515 3072 y(Banac)n(h)32 b(space)g Fz(X)7 b FA(.)52 b(F)-7 b(or)33 b(an)f(in)n(v)n(ertible)h(linear)f(op)r (erator)f Fz(J)42 b FA(w)n(e)32 b(set)p 2769 3005 55 4 v 33 w Fz(J)40 b FA(=)32 b Fy(\000)p Fz(J)3071 3041 y Fu(\000)p Fv(1)3160 3072 y FA(.)53 b(The)515 3171 y(Lipsc)n(hitz)26 b(norm)f(of)h(a)f(map)h Fz(f)35 b FA(from)25 b(a)h(metric)g(space)f Fz(M)35 b FA(to)25 b(a)h(Banac)n(h)f(space)g(is)h(de\014ned)515 3280 y(as)54 b(sup)769 3300 y Fw(m)p Fu(2)p Fw(M)961 3280 y Fy(k)p Fz(f)9 b FA(\()p Fz(m)p FA(\))p Fy(k)17 b FA(+)h(sup)1457 3300 y Fw(m)1516 3308 y Ft(1)1548 3300 y Fu(6)p Fv(=)p Fw(m)1658 3308 y Ft(2)1719 3239 y Fu(k)p Fw(f)7 b Fv(\()p Fw(m)1877 3247 y Ft(1)1909 3239 y Fv(\))p Fu(\000)p Fw(f)g Fv(\()p Fw(m)2111 3247 y Ft(2)2143 3239 y Fv(\))p Fu(k)p 1719 3260 485 4 v 1780 3308 a Fv(dist)o(\()p Fw(m)1972 3316 y Ft(1)2005 3308 y Fw(;m)2084 3316 y Ft(2)2116 3308 y Fv(\))2213 3280 y Fz(:)639 3443 y FB(Ac)m(kno)m(wledgemen)m(ts.) 86 b FA(I)45 b(thank)f(for)g(the)h(hospitalit)n(y)f(FIM)h(\(ETH,)f(Z)r (\177)-44 b(uric)n(h\),)515 3543 y(where)23 b(this)i(pap)r(er)e(w)n(as) g(completed.)36 b(The)24 b(researc)n(h)e(w)n(as)h(supp)r(orted)h(b)n(y) f(EPSR)n(C,)g(gran)n(t)515 3642 y(GR/N63055/01.)515 3917 y FC(2)134 b(Symplectic)46 b(Hilb)t(ert)g(scales)f(and)716 4066 y(Hamiltonian)i(equations)515 4265 y Fs(2.1)112 b(Hilb)s(ert)35 b(scales)j(and)g(their)e(morphisms)515 4418 y FA(Let)f Fz(X)42 b FA(b)r(e)36 b(a)g(real)e(Hilb)r(ert)j(space)d (with)i(a)g(scalar)e(pro)r(duct)h Fy(h)14 b Fr(\001)g Fz(;)27 b Fr(\001)14 b Fy(i)36 b FA(=)g Fy(h)14 b Fr(\001)g Fz(;)28 b Fr(\001)13 b Fy(i)3069 4430 y Fw(X)3168 4418 y FA(and)35 b(a)515 4528 y(Hilb)r(ert)c(basis)g Fy(f)p Fz(')1108 4540 y Fw(k)1177 4528 y Fy(j)e Fz(k)j Fy(2)1393 4507 y Fq(e)1388 4528 y Fx(Z)-6 b Fy(g)p FA(,)31 b(where)1788 4507 y Fq(e)1784 4528 y Fx(Z)24 b FA(is)31 b(a)g(coun)n(table)f(subset) h(of)g(some)g Fx(Z)3038 4498 y Fw(n)3077 4528 y FA(.)48 b(Let)31 b(us)515 4639 y(tak)n(e)22 b(a)g(p)r(ositiv)n(e)g(sequence)g Fy(f)p Fz(\022)1476 4651 y Fw(k)1539 4639 y Fy(j)h Fz(k)j Fy(2)1737 4617 y Fq(e)1733 4639 y Fx(Z)-7 b Fy(g)22 b FA(whic)n(h)g(go)r(es)g(to)g(in\014nit)n(y)h(with)g Fz(k)s FA(.)35 b(F)-7 b(or)22 b(an)n(y)g Fz(s)g FA(w)n(e)515 4749 y(de\014ne)28 b Fz(X)824 4761 y Fw(s)888 4749 y FA(as)f(a)h(Hilb)r(ert)h(space)e(with)i(the)f(Hilb)r(ert)h(basis)f Fy(f)p Fz(')2490 4761 y Fw(k)2530 4749 y Fz(\022)2571 4714 y Fu(\000)p Fw(s)2569 4774 y(k)2683 4749 y Fy(j)c Fz(k)j Fy(2)2884 4728 y Fq(e)2879 4749 y Fx(Z)-6 b Fy(g)p FA(.)38 b(By)28 b Fy(k)k Fr(\001)g Fy(k)3344 4761 y Fw(s)515 4849 y FA(and)i Fy(h)14 b Fr(\001)g Fz(;)27 b Fr(\001)14 b Fy(i)893 4861 y Fw(s)963 4849 y FA(w)n(e)35 b(denote)f(the)h(norm)f (and)h(the)g(scalar)e(pro)r(duct)h(in)h Fz(X)2794 4861 y Fw(s)2864 4849 y FA(\(in)g(particular,)515 4949 y Fz(X)584 4961 y Fv(0)648 4949 y FA(=)26 b Fz(X)36 b FA(and)30 b Fy(h)14 b Fr(\001)g Fz(;)27 b Fr(\001)14 b Fy(i)1218 4961 y Fv(0)1282 4949 y FA(=)27 b Fy(h)14 b Fr(\001)f Fz(;)28 b Fr(\001)13 b Fy(i)p FA(\).)45 b(The)30 b(totalit)n(y)f Fy(f)p Fz(X)2264 4961 y Fw(s)2299 4949 y Fy(g)h FA(is)f(called)h(a)f Fp(Hilb)l(ert)j(sc)l(ale)p FA(,)g(the)1926 5255 y(2)p eop PStoPSsaved restore %%Page: (2,3) 2 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 3 2 bop 515 523 a FA(basis)30 b Fy(f)p Fz(')818 535 y Fw(k)858 523 y Fy(g)h FA(|)f(the)h Fp(b)l(asis)j(of)f(the)g(sc)l(ale)k FA(and)31 b(the)g(scalar)e(pro)r(duct)i Fy(h)14 b Fr(\001)f Fz(;)28 b Fr(\001)13 b Fy(i)31 b FA(|)g(the)g Fp(b)l(asic)515 623 y(sc)l(alar)f(pr)l(o)l(duct)g(of)h(the)f(sc)l(ale)p FA(.)639 722 y(A)e(Hilb)r(ert)f(scale)f(ma)n(y)h(b)r(e)g(con)n(tin)n (uous)f(or)g(discrete,)h(dep)r(ending)g(on)g(whether)g Fz(s)c Fy(2)g Fx(R)515 822 y FA(or)i Fz(s)e Fy(2)h Fx(Z)o FA(.)31 b(The)26 b(ob)5 b(jects)26 b(w)n(e)h(de\014ne)f(b)r(elo)n(w)g (and)h(the)g(theorems)e(w)n(e)i(discuss)f(are)f(v)-5 b(alid)27 b(in)515 922 y(b)r(oth)h(cases.)639 1021 y(A)g(Hilb)r(ert)g (scale)f Fy(f)p Fz(X)1325 1033 y Fw(s)1360 1021 y Fy(g)g FA(p)r(ossesses)f(the)i(follo)n(wing)f(prop)r(erties:)639 1121 y(1\))h Fz(X)810 1133 y Fw(s)873 1121 y FA(is)f(compactly)g(em)n (b)r(edded)h(in)g Fz(X)1913 1133 y Fw(r)1977 1121 y FA(if)g Fz(s)23 b(>)g(r)30 b FA(and)e(is)f(dense)h(there;)639 1220 y(2\))d(the)g(spaces)f Fz(X)1199 1232 y Fw(s)1260 1220 y FA(and)g Fz(X)1487 1232 y Fu(\000)p Fw(s)1599 1220 y FA(are)g(conjugated)h(with)g(resp)r(ect)g(to)g(the)g(scalar)e (pro)r(duct)515 1320 y Fy(h)14 b Fr(\001)f Fz(;)28 b Fr(\001)13 b Fy(i)p FA(.)38 b(That)27 b(is,)h(for)f(an)n(y)g Fz(u)22 b Fy(2)i Fz(X)1601 1332 y Fw(s)1654 1320 y Fy(\\)19 b Fz(X)1797 1332 y Fv(0)1862 1320 y FA(w)n(e)27 b(ha)n(v)n(e)1063 1481 y Fy(k)p Fz(u)p Fy(k)1195 1493 y Fw(s)1252 1481 y FA(=)c(sup)p Fy(fh)p Fz(u;)14 b(u)1672 1447 y Fu(0)1694 1481 y Fy(i)23 b(j)g Fz(u)1843 1447 y Fu(0)1889 1481 y Fy(2)h Fz(X)2037 1493 y Fu(\000)p Fw(s)2142 1481 y Fy(\\)19 b Fz(X)2285 1493 y Fv(0)2322 1481 y Fz(;)28 b Fy(k)p Fz(u)2463 1447 y Fu(0)2485 1481 y Fy(k)2527 1493 y Fu(\000)p Fw(s)2637 1481 y FA(=)23 b(1)p Fy(g)p FA(;)639 1642 y(3\))31 b(the)g(norms)f Fy(k)k Fr(\001)f Fy(k)1321 1654 y Fw(s)1387 1642 y FA(satisfy)d(the)i(in)n(terp)r (olation)d(inequalit)n(y;)j(linear)e(op)r(erators)f(in)515 1742 y(the)f(spaces)e Fz(X)981 1754 y Fw(s)1044 1742 y FA(satisfy)h(the)h(in)n(terp)r(olation)f(theorem)639 1842 y(Concerning)e(these)h(and)g(other)g(prop)r(erties)f(of)h(the)h (scales)e(see)h([RS75)o(])g(and)g([Kuk00)o(].)639 1941 y(F)-7 b(or)23 b(a)f(scale)g Fy(f)p Fz(X)1153 1953 y Fw(s)1188 1941 y Fy(g)g FA(w)n(e)h(denote)g(b)n(y)f Fz(X)1812 1953 y Fu(\0001)1957 1941 y FA(and)h Fz(X)2183 1953 y Fu(1)2276 1941 y FA(the)g(linear)f(spaces)g Fz(X)2959 1953 y Fu(\0001)3104 1941 y FA(=)3192 1879 y Fq(S)3275 1941 y Fz(X)3344 1953 y Fw(s)515 2041 y FA(and)27 b Fz(X)745 2053 y Fu(1)838 2041 y FA(=)926 1979 y Fq(T)1009 2041 y Fz(X)1078 2053 y Fw(s)1113 2041 y FA(.)639 2141 y(Scales)g(of)h(Sob)r (olev)f(functions)h(are)e(the)i(most)g(imp)r(ortan)n(t)f(for)g(this)h (w)n(ork:)515 2265 y Fp(Example)33 b FA(2.1)p Fp(.)k FA(Basic)23 b(for)g(us)h(is)f(the)h(Sob)r(olev)f(scale)g(of)h (functions)f(on)h(the)g Fz(d)p FA(-dimensional)515 2364 y(torus)18 b Fy(f)p Fz(H)837 2334 y Fw(s)871 2364 y FA(\()p Fx(T)959 2334 y Fw(d)998 2364 y FA(;)c Fx(R)p FA(\))29 b(=)23 b Fz(H)1314 2334 y Fw(s)1349 2364 y FA(\()p Fx(T)1437 2334 y Fw(d)1475 2364 y FA(\))p Fy(g)p FA(.)34 b(A)19 b(space)f Fz(H)1976 2334 y Fw(s)2011 2364 y FA(\()p Fx(T)2099 2334 y Fw(d)2137 2364 y FA(\))h(is)g(formed)f(b)n(y)g(functions)h Fz(u)9 b FA(:)28 b Fx(T)3152 2334 y Fw(d)3213 2364 y Fy(!)23 b Fx(R)515 2464 y FA(suc)n(h)k(that)755 2625 y Fz(u)c FA(=)924 2546 y Fq(X)914 2730 y Fw(l)p Fu(2)p Fo(Z)1024 2714 y Fn(d)1067 2625 y Fz(u)1115 2637 y Fw(l)1140 2625 y Fz(e)1179 2591 y Fw(il)p Fu(\001)p Fw(x)1285 2625 y Fz(;)180 b Fx(C)44 b Fy(3)24 b Fz(u)1698 2637 y Fw(l)1746 2625 y FA(=)p 1833 2580 48 4 v 22 w Fz(u)1881 2637 y Fu(\000)p Fw(l)1958 2625 y Fz(;)37 b Fy(k)p Fz(u)p Fy(k)2150 2591 y Fv(2)2150 2646 y Fw(s)2209 2625 y FA(=)2297 2546 y Fq(X)2346 2725 y Fw(l)2416 2625 y FA(\(1)19 b(+)f Fy(j)p Fz(l)r Fy(j)p FA(\))2697 2591 y Fv(2)p Fw(s)2765 2625 y Fy(j)p Fz(u)2836 2637 y Fw(l)2861 2625 y Fy(j)2884 2591 y Fv(2)2945 2625 y Fz(<)k Fy(1)p Fz(:)515 2883 y FA(The)29 b(basis)g Fy(f)p Fz(')989 2895 y Fw(k)1030 2883 y Fy(g)g FA(is)h(formed)f(b)n(y)g(all)h(distinct)g(prop)r(erly)f (normalised)f(functions)i(Re)14 b Fz(e)3273 2853 y Fw(il)p Fu(\001)p Fw(x)515 2983 y FA(and)27 b(Im)14 b Fz(e)828 2953 y Fw(il)p Fu(\001)p Fw(x)934 2983 y FA(,)28 b Fz(l)c Fy(2)g Fx(Z)1174 2953 y Fw(d)1207 2983 y FA(.)639 3083 y(W)-7 b(e)29 b(shall)g(also)e(use)i(the)g(sub-scale)f Fy(f)p Fz(H)1906 3052 y Fw(s)1941 3083 y FA(\()p Fx(T)2029 3052 y Fw(d)2067 3083 y FA(\))2099 3095 y Fv(0)2137 3083 y Fy(g)p FA(,)g(where)h(a)f(space)g Fz(H)2841 3052 y Fw(s)2876 3083 y FA(\()p Fx(T)2964 3052 y Fw(d)3002 3083 y FA(\))3034 3095 y Fv(0)3101 3083 y FA(consists)515 3182 y(of)f(functions)h(from)f Fz(H)1239 3152 y Fw(s)1275 3182 y FA(\()p Fx(T)1363 3152 y Fw(d)1401 3182 y FA(\))h(with)g(zero)f (mean-v)-5 b(alue.)p 3318 3182 4 57 v 3322 3130 50 4 v 3322 3182 V 3372 3182 4 57 v 515 3306 a Fp(Example)42 b FA(2.2)p Fp(.)j FA(Consider)33 b(the)h(scale)f Fy(f)p Fz(H)1865 3276 y Fw(s)1858 3327 y Fv(0)1900 3306 y FA(\(0)p Fz(;)14 b(\031)s FA(\))p Fy(g)p FA(,)35 b(where)e(a)g(space)g Fz(H)2818 3276 y Fw(s)2811 3327 y Fv(0)2887 3306 y FA(=)g Fz(H)3061 3276 y Fw(s)3054 3327 y Fv(0)3096 3306 y FA(\(0)p Fz(;)14 b(\031)s FA(\))34 b(is)515 3406 y(formed)29 b(b)n(y)f(the)i(o)r (dd)f(2)p Fz(\031)s FA(-p)r(erio)r(dic)f(functions)i Fz(u)25 b FA(=)2184 3344 y Fq(P)2272 3364 y Fu(1)2272 3431 y Fw(k)q Fv(=1)2411 3406 y Fz(u)2459 3418 y Fw(k)2513 3406 y FA(sin)13 b Fz(k)s(x)30 b FA(suc)n(h)f(that)g Fy(k)p Fz(u)p Fy(k)3253 3376 y Fv(2)3253 3427 y Fw(s)3314 3406 y FA(=)515 3443 y Fq(P)616 3506 y Fy(j)p Fz(k)s Fy(j)708 3476 y Fv(2)p Fw(s)777 3506 y Fy(j)p Fz(u)848 3518 y Fw(k)888 3506 y Fy(j)911 3476 y Fv(2)973 3506 y Fz(<)24 b Fy(1)p FA(.)39 b(Since)28 b Fy(f)p FA(sin)13 b Fz(nx)p Fy(g)29 b FA(is)f(a)g(complete)g(system)g(of)g (eigenfunctions)g(of)h(the)515 3605 y(op)r(erator)g Fy(\000)p FA(\001)i(in)g Fz(L)1175 3617 y Fv(2)1212 3605 y FA(\(0)p Fz(;)14 b(\031)s FA(\))31 b(with)h(the)f(domain)f(of)h(de\014nition)h Fy(f)p Fz(u)27 b Fy(2)i Fz(H)2821 3575 y Fv(2)2858 3605 y FA(\(0)p Fz(;)14 b(\031)s FA(\))29 b Fy(j)g Fz(u)p FA(\(0\))f(=)515 3705 y Fz(u)p FA(\()p Fz(\031)s FA(\))23 b(=)g(0)p Fy(g)p FA(,)i(then)g(an)g(equiv)-5 b(alen)n(t)25 b(de\014nition)g(of)g(these)g(spaces)f(is)h(that)h Fz(H)2866 3675 y Fw(s)2859 3726 y Fv(0)2924 3705 y FA(=)d Fy(D)r FA(\()p Fy(\000)p FA(\001\))3276 3675 y Fw(s=)p Fv(2)515 3805 y FA(\(see)k([RS75]\).)37 b(In)28 b(particular,)598 3991 y Fz(H)674 3956 y Fv(1)667 4011 y(0)734 3991 y FA(=)22 b Fy(f)p Fz(u)h Fy(2)g Fz(H)1088 3956 y Fv(1)1125 3991 y FA(\(0)p Fz(;)14 b(\031)s FA(\))23 b Fy(j)g Fz(u)p FA(\(0\))g(=)g Fz(u)p FA(\()p Fz(\031)s FA(\))g(=)g(0)p Fy(g)p Fz(;)96 b(H)2204 3956 y Fv(2)2197 4011 y(0)2264 3991 y FA(=)22 b Fz(H)2427 3956 y Fv(2)2464 3991 y FA(\(0)p Fz(;)14 b(\031)s FA(\))19 b Fy(\\)g Fz(H)2826 3956 y Fv(1)2819 4011 y(0)2863 3991 y Fz(;)848 4126 y(H)924 4091 y Fv(3)917 4146 y(0)984 4126 y FA(=)k Fy(f)p Fz(u)f Fy(2)h Fz(H)1338 4091 y Fv(3)1375 4126 y FA(\(0)p Fz(;)14 b(\031)s FA(\))24 b Fy(j)f Fz(u)p FA(\(0\))g(=)f Fz(u)1950 4138 y Fw(xx)2029 4126 y FA(\(0\))h(=)g Fz(u)p FA(\()p Fz(\031)s FA(\))g(=)g Fz(u)2567 4138 y Fw(xx)2646 4126 y FA(\()p Fz(\031)s FA(\))h(=)f(0)p Fy(g)p Fz(:)p 3065 4126 V 3069 4073 50 4 v 3069 4126 V 3118 4126 4 57 v 229 w FA(\(2.1\))639 4287 y(Giv)n(en)g(t)n(w)n(o)g(scales)f Fy(f)p Fz(X)1366 4299 y Fw(s)1401 4287 y Fy(g)p FA(,)h Fy(f)p Fz(Y)1579 4299 y Fw(s)1615 4287 y Fy(g)f FA(and)i(a)e(linear)h (map)g Fz(L)9 b FA(:)27 b Fz(X)2493 4299 y Fu(1)2587 4287 y Fy(!)c Fz(Y)2741 4299 y Fu(\0001)2863 4287 y FA(,)h(w)n(e)f (denote)g(b)n(y)515 4386 y Fy(k)p Fz(L)p Fy(k)656 4398 y Fw(s)687 4406 y Ft(1)718 4398 y Fw(;s)769 4406 y Ft(2)829 4386 y Fy(\024)f(1)k FA(its)f(norm)g(as)g(a)g(map)g Fz(X)1771 4398 y Fw(s)1802 4406 y Ft(1)1861 4386 y Fy(!)f Fz(Y)2016 4398 y Fw(s)2047 4406 y Ft(2)2084 4386 y FA(.)36 b(W)-7 b(e)26 b(sa)n(y)e(that)h Fz(L)g FA(de\014nes)h(a)e Fp(morphism)515 4486 y(of)30 b(or)l(der)g Fz(d)d FA(of)g(the)g(t)n(w)n(o)f(scales)g (for)g Fz(s)d Fy(2)g FA([)p Fz(s)1851 4498 y Fv(0)1889 4486 y Fz(;)14 b(s)1965 4498 y Fv(1)2002 4486 y FA(],)27 b Fz(s)2114 4498 y Fv(0)2174 4486 y Fy(\024)c Fz(s)2301 4498 y Fv(1)2338 4486 y FA(,)2388 4456 y Fv(1)2452 4486 y FA(if)28 b Fy(k)p Fz(L)p Fy(k)2669 4498 y Fw(s;s)p Fu(\000)p Fw(d)2863 4486 y Fz(<)23 b Fy(1)k FA(for)f(ev)n(ery)515 4586 y Fz(s)i Fy(2)g FA([)p Fz(s)727 4598 y Fv(0)764 4586 y Fz(;)14 b(s)840 4598 y Fv(1)877 4586 y FA(].)47 b(If)31 b(in)g(addition)f(the)h(in)n(v)n(erse)e(map)i Fz(L)2155 4556 y Fu(\000)p Fv(1)2274 4586 y FA(exists)f(and)h (de\014nes)f(a)h(morphism)515 4685 y(of)i(order)g Fy(\000)p Fz(d)g FA(of)h(the)g(scales)e Fy(f)p Fz(Y)1557 4697 y Fw(s)1593 4685 y Fy(g)h FA(and)g Fy(f)p Fz(X)1946 4697 y Fw(s)1981 4685 y Fy(g)g FA(for)g Fz(s)g Fy(2)h FA([)p Fz(s)2412 4697 y Fv(0)2472 4685 y FA(+)22 b Fz(d;)14 b(s)2678 4697 y Fv(1)2737 4685 y FA(+)22 b Fz(d)p FA(],)36 b(w)n(e)d(sa)n(y)g(that)515 4785 y Fz(L)e FA(de\014nes)g(an)g Fp(isomorphism)36 b(of)e(or)l(der)g Fz(d)d FA(for)g Fz(s)f Fy(2)f FA([)p Fz(s)2234 4797 y Fv(0)2271 4785 y Fz(;)14 b(s)2347 4797 y Fv(1)2385 4785 y FA(].)48 b(If)32 b Fy(f)p Fz(X)2677 4797 y Fw(s)2711 4785 y Fy(g)d FA(=)g Fy(f)p Fz(Y)2966 4797 y Fw(s)3002 4785 y Fy(g)p FA(,)i(then)h(an)515 4885 y(isomorphism)26 b(is)i(called)f(an)g Fp(automorphism)p FA(.)p 515 4929 1146 4 v 607 4983 a Fm(1)642 5006 y Fl(or)c Fk(s)c Fj(2)h Fl(\()p Fk(s)907 5015 y Fm(0)942 5006 y Fk(;)11 b(s)1006 5015 y Fm(1)1040 5006 y Fl(\),)24 b(etc.)1926 5255 y FA(3)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 4 3 bop 515 523 a Fp(Example)40 b FA(2.3)p Fp(.)j FA(Multiplication)32 b(b)n(y)f(a)g(non-v)-5 b(anishing)30 b Fz(C)2371 493 y Fw(r)2408 523 y FA(-smo)r(oth)g(function)i(de\014nes)f(a)515 623 y(zero-order)24 b(automorphism)j(of)h(the)g(Sob)r(olev)f(scale)f Fy(f)p Fz(H)2320 593 y Fw(s)2355 623 y FA(\()p Fx(T)2443 593 y Fw(n)2488 623 y FA(\))p Fy(g)h FA(for)h Fy(\000)p Fz(r)d Fy(\024)e Fz(s)g Fy(\024)f Fz(r)r FA(.)p 3318 623 4 57 v 3322 570 50 4 v 3322 623 V 3372 623 4 57 v 639 756 a(If)f Fz(L)f FA(is)g(a)g(morphism)g(of)g(scales)g Fy(f)p Fz(X)1734 768 y Fw(s)1768 756 y Fy(g)p FA(,)i Fy(f)p Fz(Y)1945 768 y Fw(s)1980 756 y Fy(g)e FA(of)g(order)f Fz(d)i FA(for)f Fz(s)j Fy(2)g FA([)p Fz(s)2725 768 y Fv(0)2762 756 y Fz(;)14 b(s)2838 768 y Fv(1)2875 756 y FA(],)22 b(then)f(adjoin)n(t)515 855 y(maps)27 b Fz(L)789 825 y Fu(\003)854 855 y FA(form)g(a)g(morphism)g(of)h(the)g(scales)e Fy(f)p Fz(Y)2066 867 y Fw(s)2101 855 y Fy(g)h FA(and)h Fy(f)p Fz(X)2443 867 y Fw(s)2477 855 y Fy(g)f FA(of)h(the)g(same)f (order)f Fz(d)i FA(for)515 955 y Fz(s)23 b Fy(2)g FA([)p Fy(\000)p Fz(s)782 967 y Fv(1)837 955 y FA(+)18 b Fz(d;)c Fy(\000)p Fz(s)1104 967 y Fv(0)1160 955 y FA(+)k Fz(d)p FA(].)37 b(It)28 b(is)f(called)h(the)g Fp(adjoint)j(morphism)p FA(.)639 1054 y(If)38 b Fz(L)h FA(=)f Fz(L)988 1024 y Fu(\003)1063 1054 y FA(\()p Fz(L)h FA(=)g Fy(\000)p Fz(L)1417 1024 y Fu(\003)1454 1054 y FA(\))f(on)f(the)h(space)e Fz(X)2102 1066 y Fu(1)2172 1054 y FA(,)k(then)e(the)g(morphism)e Fz(L)h FA(is)g(called)515 1154 y(symmetric)27 b(\(an)n(tisymmetric\).) 639 1254 y(If)j Fz(L)e FA(is)h(a)g(symmetric)f(morphism)h(of)g Fy(f)p Fz(X)1967 1266 y Fw(s)2001 1254 y Fy(g)g FA(of)g(order)f Fz(d)h FA(for)f Fz(s)e Fy(2)f FA([)p Fz(s)2794 1266 y Fv(0)2832 1254 y Fz(;)14 b(d)19 b Fy(\000)g Fz(s)3054 1266 y Fv(0)3091 1254 y FA(],)30 b(where)515 1353 y Fz(s)554 1365 y Fv(0)632 1353 y Fy(\025)41 b Fz(d=)p FA(2,)f(then)f(the)g (adjoin)n(t)f(morphism)g Fz(L)2030 1323 y Fu(\003)2107 1353 y FA(is)g(de\014ned)h(for)f Fz(s)j Fy(2)g FA([)p Fz(s)2874 1365 y Fv(0)2911 1353 y Fz(;)14 b(d)26 b Fy(\000)f Fz(s)3146 1365 y Fv(0)3184 1353 y FA(])38 b(and)515 1453 y(coincide)e(with)h Fz(L)f FA(on)h Fz(X)1327 1465 y Fu(1)1397 1453 y FA(;)k(hence,)e Fz(L)1783 1423 y Fu(\003)1859 1453 y FA(=)e Fz(L)p FA(.)64 b(W)-7 b(e)37 b(call)f Fz(L)g FA(a)g Fp(selfadjoint)k(morphism)p FA(.)515 1553 y(An)n(ti-selfadjoin)n (t)27 b(morphisms)g(are)g(de\014ned)h(similarly)-7 b(.)515 1685 y Fp(Example)39 b FA(2.4)p Fp(.)j FA(The)30 b(op)r(erator)d(\001)j (de\014nes)g(a)f(selfadjoin)n(t)h(morphism)f(of)h(order)e(2)h(of)h(the) 515 1785 y(Sob)r(olev)h(scale)h Fy(f)p Fz(H)1150 1755 y Fw(s)1184 1785 y FA(\()p Fx(T)1272 1755 y Fw(n)1317 1785 y FA(\))p Fy(g)g FA(for)g Fy(\0001)e Fz(<)g(s)h(<)f Fy(1)p FA(.)50 b(The)32 b(op)r(erators)e Fz(@)5 b(=@)g(x)2883 1797 y Fw(j)2918 1785 y FA(,)33 b(1)d Fy(\024)g Fz(j)36 b Fy(\024)30 b Fz(n)p FA(,)515 1885 y(de\014ne)e(an)n(ti-selfadjoin)n (t)f(morphisms)h(of)g(order)e(one.)38 b(The)29 b(automorphism)e(in)h (Example)515 1984 y(1.1)e(is)i(selfadjoin)n(t.)p 3318 1984 V 3322 1932 50 4 v 3322 1984 V 3372 1984 4 57 v 639 2117 a(Let)g Fy(f)p Fz(Y)878 2129 y Fw(s)914 2117 y Fy(g)p FA(,)f Fy(f)p Fz(Y)1096 2129 y Fw(s)1132 2117 y Fy(g)g FA(b)r(e)i(t)n(w)n(o)e(scales)g(and)h Fz(O)1929 2129 y Fw(s)1988 2117 y Fy(\032)c Fz(X)2146 2129 y Fw(s)2181 2117 y FA(,)k Fz(s)c Fy(2)g FA([)p Fz(a;)14 b(b)p FA(],)28 b(b)r(e)g(a)g(system)f(of)h(\(op)r(en\))515 2217 y(domains,)f (compatible)g(in)h(the)g(follo)n(wing)e(sense:)1269 2399 y Fz(O)1332 2411 y Fw(s)1363 2419 y Ft(1)1418 2399 y Fy(\\)19 b Fz(O)1555 2411 y Fw(s)1586 2419 y Ft(2)1646 2399 y FA(=)k Fz(O)1797 2411 y Fw(s)1828 2419 y Ft(2)1948 2399 y FA(if)42 b Fz(a)23 b Fy(\024)g Fz(s)2232 2411 y Fv(1)2292 2399 y Fy(\024)g Fz(s)2419 2411 y Fv(2)2479 2399 y Fy(\024)f Fz(b:)515 2582 y FA(Let)36 b Fz(F)21 b FA(:)31 b Fz(O)863 2594 y Fw(a)940 2582 y Fy(!)38 b Fz(Y)1109 2594 y Fu(\0001)1267 2582 y FA(b)r(e)f(a)f(map)g(suc)n(h)f (that)i(for)e(ev)n(ery)g Fz(s)i Fy(2)h FA([)p Fz(a;)14 b(b)p FA(])36 b(its)g(restriction)f(to)515 2682 y Fz(O)578 2694 y Fw(s)645 2682 y FA(de\014nes)c(an)g(analytic)f(\()p Fz(C)1456 2651 y Fw(k)1498 2682 y FA(-smo)r(oth\))h(map)g Fz(F)21 b FA(:)29 b Fz(O)2232 2694 y Fw(s)2296 2682 y Fy(!)g Fz(Y)2456 2694 y Fw(s)p Fu(\000)p Fw(d)2579 2682 y FA(.)47 b(Then)31 b Fz(F)44 b FA(is)31 b(called)f(an)515 2781 y(analytic)d(\()p Fz(C)928 2751 y Fw(k)969 2781 y FA(-smo)r(oth\))g(morphism)h(of)f(order)f Fz(d)i FA(for)f Fz(s)c Fy(2)h FA([)p Fz(a;)14 b(b)p FA(].)515 2914 y Fp(Example)34 b FA(2.5)p Fp(.)k FA(Let)24 b Fy(f)p Fz(X)1281 2926 y Fw(s)1316 2914 y Fy(g)g FA(b)r(e)g(the)h(Sob)r(olev)f(scale)f Fy(f)p Fz(H)2250 2884 y Fw(s)2285 2914 y FA(\()p Fx(T)2373 2884 y Fw(d)2411 2914 y FA(\))p Fy(g)h FA(and)g Fz(f)9 b FA(\()p Fz(u;)14 b(x)p FA(\))25 b(b)r(e)f(a)g(smo)r(oth)515 3014 y(function.)43 b(Then)30 b(the)g(map)f Fz(F)39 b FA(:)26 b Fz(u)p FA(\()p Fz(x)p FA(\))h Fy(7!)g Fz(f)9 b FA(\()p Fz(u)p FA(\()p Fz(x)p FA(\))p Fz(;)14 b(x)p FA(\),)31 b Fz(X)2345 3026 y Fw(a)2411 3014 y Fy(!)c Fz(X)2590 3026 y Fw(a)2630 3014 y FA(,)j(is)f(smo)r(oth)h(if)g Fz(a)c(>)3313 2981 y Fv(1)p 3313 2995 34 4 v 3313 3042 a(2)3356 3014 y FA(,)515 3113 y(so)h(on)g(these)h(spaces)e(ord)14 b Fz(F)34 b FA(=)23 b(0.)36 b(If)28 b Fz(f)37 b FA(is)27 b(analytic,)g(then)h(so)f(is)h Fz(F)12 b FA(.)639 3213 y(No)n(w)25 b(let)h(us)f(assume)g(that)h Fz(d)d FA(=)g(1,)i Fz(f)34 b FA(is)26 b(analytic,)f Fz(f)9 b FA(\(0)p Fz(;)14 b(x)p FA(\))23 b Fy(\021)g FA(0)i(and)g(consider)f Fz(F)38 b FA(as)24 b(a)515 3313 y(map)d(in)h(the)g(scale)f Fy(f)p Fz(H)1232 3282 y Fw(s)1225 3333 y Fv(0)1290 3313 y FA(=)i Fz(H)1454 3282 y Fw(s)1447 3333 y Fv(0)1489 3313 y FA(\(0)p Fz(;)14 b(\031)s FA(\))p Fz(;)28 b(s)23 b Fy(2)g Fx(Z)p Fy(g)o FA(.)29 b(F)-7 b(or)21 b Fz(s)i Fy(\025)g FA(1)e(the)h(map)g Fz(F)35 b FA(:)23 b Fz(H)2910 3282 y Fw(s)2903 3333 y Fv(0)2968 3313 y Fy(!)g Fz(H)3150 3282 y Fw(s)3186 3313 y FA(\(0)p Fz(;)14 b(\031)s FA(\))515 3412 y(is)31 b(analytic.)49 b(Since)32 b Fz(F)12 b(u)p FA(\(0\))30 b(=)g Fz(F)12 b(u)p FA(\()p Fz(\031)s FA(\))30 b(=)g(0,)i(then)h(due)f(to)f(\(2.1\))h (for)f Fz(s)f FA(=)g(1)h(and)h Fz(s)e FA(=)f(2)515 3512 y Fz(F)12 b FA(\()p Fz(H)688 3482 y Fw(s)681 3532 y Fv(0)723 3512 y FA(\))24 b Fy(\032)e Fz(H)942 3482 y Fw(s)935 3532 y Fv(0)977 3512 y FA(.)37 b(So)25 b(on)g(the)g(spaces)g Fz(H)1732 3482 y Fv(1)1725 3532 y(0)1794 3512 y FA(and)g Fz(H)2029 3482 y Fv(2)2022 3532 y(0)2092 3512 y FA(w)n(e)g(ha)n(v)n(e)f (ord)14 b Fz(F)34 b FA(=)23 b(0.)35 b(Since)26 b(in)f(general)515 3611 y(for)j Fz(u)23 b Fy(2)i Fz(H)870 3581 y Fu(1)863 3632 y Fv(0)940 3611 y FA(,)k Fz(F)12 b FA(\()p Fz(u)p FA(\))24 b Fy(2)g Fz(H)1348 3581 y Fv(2)1341 3632 y(0)1414 3611 y FA(but)38 b Fz(=)-51 b Fy(2)24 b Fz(H)1722 3581 y Fv(3)1715 3632 y(0)1788 3611 y FA(\(see)k(\(2.1\)\),)h(then)f(on)g (the)h(spaces)e Fz(H)2990 3581 y Fw(s)2983 3632 y Fv(0)3026 3611 y Fz(;)14 b(s)24 b Fy(\025)g FA(3)j(w)n(e)515 3711 y(ha)n(v)n(e)f(ord)14 b Fz(F)34 b(>)23 b FA(0.)639 3811 y(If)33 b Fz(f)9 b FA(\()p Fz(u;)14 b(x)p FA(\))32 b(is)g(o)r(dd)h(in)f Fz(u)g FA(and)g(ev)n(en)f(in)i Fz(x)g FA(\(e.g.,)g(is)f Fz(x)p FA(-indep)r(enden)n(t\),)j(or)c(vice)h(v)n(ersa,)515 3910 y(then)c Fz(F)12 b FA(\()p Fz(H)877 3880 y Fw(s)870 3931 y Fv(0)912 3910 y FA(\))24 b Fy(\032)e Fz(H)1131 3880 y Fw(s)1124 3931 y Fv(0)1194 3910 y FA(for)27 b Fz(s)c Fy(\025)g FA(1,)k(so)g(ord)14 b Fz(F)34 b FA(=)23 b(0)k(for)g(an)n(y)g Fz(s)c Fy(\025)f FA(1.)p 3318 3910 4 57 v 3322 3858 50 4 v 3322 3910 V 3372 3910 4 57 v 639 4043 a(Giv)n(en)38 b(a)g Fz(C)1035 4013 y Fw(k)1076 4043 y FA(-smo)r(oth)g(function)h Fz(H)16 b FA(:)31 b Fz(X)1952 4055 y Fw(d)2031 4043 y Fy(\033)40 b Fz(O)2199 4055 y Fw(d)2279 4043 y Fy(!)h Fx(R)p FA(,)47 b Fz(k)c Fy(\025)d FA(1,)h(w)n(e)d(consider)f(its)515 4143 y Fp(gr)l(adient)30 b(map)k FA(with)28 b(resp)r(ect)f(to)h(the)g(paring)e Fy(h)14 b Fr(\001)g Fz(;)27 b Fr(\001)14 b Fy(i)p FA(:)971 4325 y Fy(r)p Fz(H)i FA(:)27 b Fz(O)1238 4337 y Fw(d)1301 4325 y Fy(!)c Fz(X)1476 4337 y Fu(\000)p Fw(d)1566 4325 y Fz(;)180 b Fy(hr)p Fz(H)7 b FA(\()p Fz(u)p FA(\))p Fz(;)14 b(v)s Fy(i)24 b FA(=)e Fz(dH)7 b FA(\()p Fz(u)p FA(\))p Fz(v)50 b Fy(8)p Fz(v)25 b Fy(2)f Fz(X)2862 4337 y Fw(d)2900 4325 y Fz(:)515 4508 y FA(The)j(map)h Fy(r)p Fz(H)35 b FA(is)27 b Fz(C)1191 4478 y Fw(k)q Fu(\000)p Fv(1)1317 4508 y FA(-smo)r(oth.)639 4608 y(If)35 b Fz(O)792 4620 y Fw(d)864 4608 y FA(b)r(elongs)f(to)f(a)h(system)f(of)h (compatible)g(domains)f Fz(O)2562 4620 y Fw(s)2598 4608 y FA(,)i Fz(a)e Fy(\024)h Fz(s)f Fy(\024)g Fz(b)p FA(,)i(and)f(the)515 4707 y(gradien)n(t)26 b(map)i Fy(r)p Fz(H)35 b FA(de\014nes)28 b(a)f Fz(C)1605 4677 y Fw(k)q Fu(\000)p Fv(1)1731 4707 y FA(-smo)r(oth)h(morphism)f(of)h(order)e Fz(d)2796 4719 y Fw(H)2887 4707 y FA(for)i Fz(a)23 b Fy(\024)g Fz(s)g Fy(\024)g Fz(b)p FA(,)515 4807 y(w)n(e)k(write)g(that)h(ord)13 b Fy(r)p Fz(H)30 b FA(=)23 b Fz(d)1462 4819 y Fw(H)1525 4807 y FA(.)1926 5255 y(4)p eop PStoPSsaved restore %%Page: (4,5) 3 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 5 4 bop 515 523 a Fs(2.2)112 b(Symplectic)35 b(structures)515 676 y FA(F)-7 b(or)22 b(simplicit)n(y)g(w)n(e)g(restrict)g(ourselv)n (es)f(to)h(constan)n(t-co)r(e\016cien)n(t)f(symplectic)i(structures.) 515 776 y(F)-7 b(or)27 b(the)h(general)e(case)h(see)g([Kuk00)n(].)639 876 y(Let)32 b Fy(f)p Fz(X)903 888 y Fw(s)938 876 y Fy(g)g FA(b)r(e)g(a)g(Hilb)r(ert)g(scale)g(and)f Fz(J)40 b FA(b)r(e)33 b(its)f(an)n(ti-selfadjoin)n(t)f(automorphism)g(of)515 975 y(order)37 b Fz(d)i FA(for)g Fy(\0001)i Fz(<)g(s)h(<)f Fy(1)p FA(.)71 b(Then)39 b(the)g(op)r(erator)p 2351 908 55 4 v 37 w Fz(J)50 b FA(=)42 b Fy(\000)p Fz(J)2673 945 y Fu(\000)p Fv(1)2800 975 y FA(de\014nes)d(an)g(an)n(ti-)515 1075 y(selfadjoin)n(t)27 b(automorphism)g(of)g(order)g Fy(\000)p Fz(d)p FA(.)36 b(W)-7 b(e)28 b(de\014ne)g(a)f(t)n(w)n(o-form) f Fz(\013)2801 1087 y Fv(2)2866 1075 y FA(as)1664 1247 y Fz(\013)1717 1259 y Fv(2)1778 1247 y FA(=)p 1865 1181 V 22 w Fz(J)c(dx)d Fy(^)g Fz(dx;)515 1420 y FA(where)30 b(b)n(y)h(de\014nition)p 1249 1353 V 31 w Fz(J)22 b(dx)f Fy(^)h Fz(dx)14 b FA([)p Fz(\030)t(;)g(\021)s FA(])29 b(=)f Fy(h)p 1930 1353 V Fz(J)9 b(\030)t(;)14 b(\021)s Fy(i)p Fz(:)31 b FA(Clearly)-7 b(,)p 2501 1353 V 31 w Fz(J)22 b(dx)f Fy(^)h Fz(dx)31 b FA(de\014nes)g(a)g(con-)515 1519 y(tin)n(uous)e(sk)n(ew-symmetric)f(bilinear)i(form)f(on)g Fz(X)2102 1531 y Fw(r)2159 1519 y Fy(\002)19 b Fz(X)2312 1531 y Fw(r)2379 1519 y FA(if)30 b Fz(r)f Fy(\025)d(\000)p Fz(d=)p FA(2.)42 b(Therefore)29 b(an)n(y)515 1619 y(space)c Fz(X)804 1631 y Fw(r)841 1619 y FA(,)h Fz(r)g Fy(\025)d(\000)p Fz(d=)p FA(2,)i(b)r(ecomes)h(a)g Fp(symple)l(ctic)32 b FA(\()p Fp(Hilb)l(ert)8 b FA(\))27 b Fp(sp)l(ac)l(e)33 b FA(and)26 b(w)n(e)g(shall)g(write)g(it)515 1719 y(as)h(a)g(pair)g(\() p Fz(X)958 1731 y Fw(r)995 1719 y Fz(;)14 b(\013)1085 1731 y Fv(2)1122 1719 y FA(\).)639 1818 y(The)28 b(pair)f(\()p Fy(f)p Fz(X)1124 1830 y Fw(s)1159 1818 y Fy(g)p Fz(;)14 b(\013)1291 1830 y Fv(2)1328 1818 y FA(\))28 b(is)f(called)g(a)h Fp(symple)l(ctic)33 b FA(\()p Fp(Hilb)l(ert)8 b FA(\))28 b Fp(sc)l(ale)p FA(.)515 1947 y Fp(Example)41 b FA(2.6)p Fp(.)j FA(Let)33 b(us)f(tak)n(e)g(the)g(index-set)h Fy(Z)39 b FA(to)32 b(b)r(e)h(the)g(union)f(of)g(non-in)n(tersecting)515 2047 y(subsets)26 b Fy(Z)861 2059 y Fv(+)943 2047 y FA(and)g Fy(Z)1163 2059 y Fu(\000)1219 2047 y FA(,)h(pro)n(vided)e(with)i(an)f (in)n(v)n(olution)g Fy(Z)k(!)23 b(Z)33 b FA(whic)n(h)27 b(will)f(b)r(e)h(denoted)515 2146 y Fz(j)38 b Fy(7!)33 b(\000)p Fz(j)5 b FA(,)35 b(suc)n(h)f(that)g Fy(\000Z)1370 2158 y Fu(\006)1459 2146 y FA(=)f Fy(Z)1617 2158 y Fu(\007)1673 2146 y FA(.)55 b(Let)34 b(us)g(consider)f(a)g(Hilb)r(ert)h(scale)f Fy(f)p Fz(X)3032 2158 y Fw(s)3067 2146 y Fy(g)g FA(with)h(a)515 2246 y(basis)29 b Fy(f)p Fz(\036)812 2258 y Fw(k)853 2246 y FA(,)i Fz(k)f Fy(2)e(Z)7 b(g)29 b FA(and)h(a)g(sequence)f Fy(f)p Fz(\022)1864 2258 y Fw(k)1905 2246 y Fz(;)14 b(k)30 b Fy(2)d(Z)7 b(g)p FA(,)30 b(suc)n(h)g(that)g Fz(\022)2670 2258 y Fu(\000)p Fw(j)2785 2246 y Fy(\021)c Fz(\022)2915 2258 y Fw(j)2950 2246 y FA(.)45 b(T)-7 b(ak)n(e)29 b Fz(J)38 b FA(to)515 2346 y(b)r(e)28 b(the)g(linear)f(op)r(erator,)e (de\014ned)j(b)n(y)g(the)g(relations)1033 2518 y Fz(J)8 b(\036)1136 2530 y Fw(k)1200 2518 y FA(=)23 b Fz(\036)1337 2530 y Fu(\000)p Fw(k)1513 2518 y Fy(8)14 b Fz(k)25 b Fy(2)e(Z)1780 2530 y Fv(+)1850 2518 y Fz(;)96 b(J)8 b(\036)2072 2530 y Fw(k)2137 2518 y FA(=)22 b Fy(\000)p Fz(\036)2338 2530 y Fu(\000)p Fw(k)2514 2518 y Fy(8)14 b Fz(k)25 b Fy(2)f(Z)2782 2530 y Fu(\000)2838 2518 y Fz(:)515 2691 y FA(It)d(de\014nes)g(an)f(an)n(ti-selfadjoin)n(t)g(automorphism)g(of)g (the)h(scale)f(of)h(zero)f(order,)h(and)p 3137 2624 V 20 w Fz(J)31 b FA(=)23 b Fz(J)8 b FA(.)515 2790 y(The)23 b(symplectic)g(scale)g(\()p Fy(f)p Fz(X)1423 2802 y Fw(s)1458 2790 y Fy(g)p Fz(;)14 b(\013)1590 2802 y Fv(2)1649 2790 y FA(=)p 1737 2724 V 23 w Fz(J)8 b(dx)i Fy(^)g Fz(dx)24 b FA(=)f Fz(J)8 b(dx)i Fy(^)g Fz(dx)p FA(\))24 b(will)f(b)r(e)h(called) f(a)f Fp(Darb)l(oux)515 2890 y(sc)l(ale)p FA(.)p 3318 2890 4 57 v 3322 2837 50 4 v 3322 2890 V 3372 2890 4 57 v 639 3019 a(Let)34 b(\()p Fy(f)p Fz(X)937 3031 y Fw(s)972 3019 y Fy(g)p Fz(;)14 b(\013)1104 3031 y Fv(2)1173 3019 y FA(=)p 1269 2952 55 4 v 31 w Fz(J)22 b(dx)h Fy(^)g Fz(dx)p FA(\))34 b(and)f(\()p Fy(f)p Fz(Y)1973 3031 y Fw(s)2008 3019 y Fy(g)p Fz(;)14 b(\014)2134 3031 y Fv(2)2203 3019 y FA(=)p 2300 2952 65 4 v 32 w(\007)f Fz(dy)25 b Fy(^)e Fz(dy)s FA(\))33 b(b)r(e)h(t)n(w)n(o)e(symplectic)515 3118 y(Hilb)r(ert)d(scales)f(and)h Fz(O)1261 3130 y Fw(s)1322 3118 y Fy(\032)c Fz(X)1481 3130 y Fw(s)1516 3118 y FA(,)k Fz(a)d Fy(\024)e Fz(s)i Fy(\024)e Fz(b)p FA(,)29 b(b)r(e)h(a)e(system)h (of)g(compatible)g(domains.)40 b(A)515 3218 y Fz(C)580 3188 y Fv(1)617 3218 y FA(-smo)r(oth)27 b(morphism)g(of)h(order)e Fz(d)1681 3230 y Fv(1)1386 3390 y Fz(F)21 b FA(:)28 b Fz(O)1574 3402 y Fw(s)1633 3390 y Fy(!)23 b Fz(Y)1787 3402 y Fw(s)p Fu(\000)p Fw(d)1905 3410 y Ft(1)1942 3390 y Fz(;)180 b(a)23 b Fy(\024)f Fz(s)h Fy(\024)g Fz(b;)515 3563 y FA(is)k Fp(symple)l(ctic)34 b FA(if)28 b Fz(F)1141 3533 y Fu(\003)1179 3563 y Fz(\014)1226 3575 y Fv(2)1286 3563 y FA(=)23 b Fz(\013)1427 3575 y Fv(2)1464 3563 y FA(.)37 b(That)28 b(is,)f(if)h Fy(h)p 1946 3496 V FA(\007)p Fz(F)2064 3575 y Fu(\003)2103 3563 y FA(\()p Fz(x)p FA(\))p Fz(\030)t(;)14 b(F)2344 3575 y Fu(\003)2383 3563 y FA(\()p Fz(x)p FA(\))p Fz(\021)s Fy(i)2570 3575 y Fw(Y)2652 3563 y Fy(\021)23 b(h)p 2772 3496 55 4 v Fz(J)8 b(\030)t(;)14 b(\021)s Fy(i)2979 3575 y Fw(X)3043 3563 y FA(,)28 b(or)1523 3735 y Fz(F)1588 3701 y Fu(\003)1626 3735 y FA(\()p Fz(x)p FA(\))p 1737 3669 65 4 v(\007)q Fz(F)1856 3747 y Fu(\003)1894 3735 y FA(\()p Fz(x)p FA(\))c(=)p 2117 3669 55 4 v 23 w Fz(J)91 b Fy(8)p Fz(x:)515 3908 y FA(A)38 b(symplectic)g(morphism)g Fz(F)50 b FA(as)37 b(ab)r(o)n(v)n(e)g(is)h(called)g(a)f Fp(symple)l(ctomorphism)47 b FA(if)39 b(it)g(is)e(a)515 4008 y(di\013eomorphism.)515 4238 y Fs(2.3)112 b(Hamiltonian)35 b(equations)515 4391 y FA(T)-7 b(o)30 b(a)h Fz(C)778 4361 y Fv(1)816 4391 y FA(-smo)r(oth)f(function)i Fz(h)f FA(on)g(a)f(domain)h Fz(O)2101 4403 y Fw(d)2169 4391 y Fy(\032)d Fz(X)2331 4403 y Fw(d)2370 4391 y FA(,)k(the)f(symplectic)g (form)g Fz(\013)3236 4403 y Fv(2)3305 4391 y FA(as)515 4491 y(ab)r(o)n(v)n(e)20 b(corresp)r(onds)g(the)j Fp(Hamiltonian)i(ve)l (ctor)g(\014eld)31 b Fz(V)2273 4503 y Fw(h)2317 4491 y FA(,)23 b(de\014ned)f(b)n(y)g(the)g(usual)g(relation)515 4591 y(\(cf.)28 b([Arn89)o(,)g(HZ94)o(]\):)1433 4690 y Fz(\013)1486 4702 y Fv(2)1523 4690 y FA([)p Fz(V)1594 4702 y Fw(h)1638 4690 y FA(\()p Fz(x)p FA(\))p Fz(;)14 b(\030)t FA(])24 b(=)f Fy(\000)p Fz(dh)p FA(\()p Fz(x)p FA(\))p Fz(\030)87 b Fy(8)p Fz(\030)t(:)515 4834 y FA(That)27 b(is,)h Fy(h)p 861 4767 V Fz(J)8 b(V)963 4846 y Fw(h)1007 4834 y FA(\()p Fz(x)p FA(\))p Fz(;)14 b(\030)t Fy(i)24 b(\021)f(\000hr)p Fz(h)p FA(\()p Fz(x)p FA(\))p Fz(;)14 b(\030)t Fy(i)29 b FA(and)1637 5006 y Fz(V)1685 5018 y Fw(h)1728 5006 y FA(\()p Fz(x)p FA(\))24 b(=)f Fz(J)8 b Fy(r)p Fz(h)p FA(\()p Fz(x)p FA(\))p Fz(:)1926 5255 y FA(5)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 6 5 bop 515 523 a FA(The)30 b(v)n(ector)e(\014eld)i Fz(V)1169 535 y Fw(h)1243 523 y FA(de\014nes)g(a)f(con)n(tin)n(uous)g(map)h Fz(O)2257 535 y Fw(d)2323 523 y Fy(!)d Fz(X)2502 535 y Fu(\000)p Fw(d)p Fu(\000)p Fw(d)2676 543 y Fn(J)2719 523 y FA(.)43 b(Usually)30 b(w)n(e)f(shall)515 623 y(assume)d(that)h Fz(V)1028 635 y Fw(h)1098 623 y FA(is)g(smo)r(other)f(than)h(that)g (and)g(de\014nes)f(a)h(smo)r(oth)f(morphism)h(of)f(order)515 722 y Fz(d)558 734 y Fv(1)618 722 y Fy(\024)d FA(2)p Fz(d)18 b FA(+)g Fz(d)935 734 y Fw(J)1009 722 y FA(for)27 b(all)h Fz(s)f FA(from)h(some)f(segmen)n(t.)639 822 y(F)-7 b(or)27 b(an)n(y)g Fz(C)1010 792 y Fv(1)1047 822 y FA(-smo)r(oth)g (function)h Fz(h)g FA(on)f Fz(O)1947 834 y Fw(d)2004 822 y Fy(\002)18 b Fx(R)33 b FA(w)n(e)27 b(denote)h(b)n(y)f Fz(V)2727 834 y Fw(h)2798 822 y FA(the)h(non-autono-)515 922 y(mous)36 b(v)n(ector)f(\014eld)i Fz(V)1236 934 y Fw(h)1280 922 y FA(\()p Fz(x;)14 b(t)p FA(\))39 b(=)f Fz(J)8 b Fy(r)1723 934 y Fw(x)1765 922 y Fz(h)p FA(\()p Fz(x;)14 b(t)p FA(\),)40 b(where)c Fy(r)2372 934 y Fw(x)2451 922 y FA(is)g(the)h(gradien)n(t)f(in)h Fz(x)p FA(,)i(and)515 1021 y(consider)26 b(the)i(corresp)r(onding)e Fp(Hamiltonian)31 b(e)l(quation)j FA(\(or)27 b Fp(Hamiltonian)k(system)6 b FA(\))1484 1177 y(_)-37 b Fz(x)23 b FA(=)g Fz(J)8 b Fy(r)1751 1189 y Fw(x)1793 1177 y Fz(h)p FA(\()p Fz(x;)14 b(t)p FA(\))24 b(=)e Fz(V)2178 1189 y Fw(h)2222 1177 y FA(\()p Fz(x;)14 b(t)p FA(\))p Fz(:)785 b FA(\(2.2\))639 1333 y(A)27 b(partial)f(di\013eren)n(tial)g(equation,)g(supplemen)n (ted)h(b)n(y)f(some)g(b)r(oundary)f(conditions,)515 1433 y(is)36 b(called)g(a)g Fp(Hamiltonian)j(p)l(artial)g(di\013er)l(ential) g(e)l(quation)p FA(,)g(or)d(an)g Fp(HPDE)p FA(,)g(if)h(under)f(a)515 1532 y(suitable)25 b(c)n(hoice)f(of)i(a)e(symplectic)i(Hilb)r(ert)g (scale)e(\()p Fy(f)p Fz(X)2255 1544 y Fw(s)2290 1532 y Fy(g)p Fz(;)14 b(\013)2422 1544 y Fv(2)2459 1532 y FA(\),)26 b(a)f(domain)g Fz(O)2963 1544 y Fw(d)3025 1532 y Fy(\032)d Fz(X)3181 1544 y Fw(d)3245 1532 y FA(and)515 1632 y(a)33 b(Hamiltonian)h Fz(h)p FA(,)h(it)f(can)g(b)r(e)g(written)g (in)g(the)g(form)g(\(2.2\))o(.)56 b(In)34 b(this)g(case)f(the)h(v)n (ector)515 1731 y(\014eld)28 b Fz(V)743 1743 y Fw(h)814 1731 y FA(is)f(un)n(b)r(ounded,)h(ord)13 b Fz(V)1531 1743 y Fw(h)1598 1731 y FA(=)22 b Fz(d)1728 1743 y Fv(1)1789 1731 y Fz(>)h FA(0.)36 b(That)28 b(is,)1550 1887 y Fz(V)1598 1899 y Fw(h)1651 1887 y FA(:)g Fz(O)1765 1899 y Fw(d)1822 1887 y Fy(\002)18 b Fx(R)29 b Fy(!)23 b Fz(X)2163 1899 y Fw(d)p Fu(\000)p Fw(d)2285 1907 y Ft(1)2321 1887 y Fz(:)515 2043 y FA(Usually)29 b Fz(O)880 2055 y Fw(d)948 2043 y FA(b)r(elongs)f(to)h(a)g(system)g(of)g(compatible)g(domains)f Fz(O)2612 2055 y Fw(s)2648 2043 y FA(,)i Fz(s)25 b Fy(\025)h Fz(d)2899 2055 y Fv(0)2936 2043 y FA(,)k(and)f Fz(V)3200 2055 y Fw(h)3272 2043 y FA(\(as)515 2143 y(a)e(function)h(of)g Fz(x)p FA(\))g(de\014nes)g(an)f(analytic)g(morphism)g(of)h(order)e Fz(d)2558 2155 y Fv(1)2623 2143 y FA(for)h Fz(s)c Fy(\025)g Fz(d)2943 2155 y Fv(0)2980 2143 y FA(.)639 2243 y(A)k(con)n(tin)n(uous) e(curv)n(e)h Fz(x)9 b FA(:)28 b([)p Fz(t)1523 2255 y Fv(0)1560 2243 y Fz(;)14 b(t)1627 2255 y Fv(1)1664 2243 y FA(])23 b Fy(!)h Fz(O)1880 2255 y Fw(d)1945 2243 y FA(is)i(called)g(a)g Fp(solution)j(of)47 b FA(\(2.2\))28 b Fp(in)h(the)f(sp)l(ac)l(e)515 2342 y Fz(X)584 2354 y Fw(d)654 2342 y FA(if)33 b(it)f(de\014nes)h(a)e Fz(C)1238 2312 y Fv(1)1276 2342 y FA(-smo)r(oth)g(map)h Fz(x)9 b FA(:)30 b([)p Fz(t)1952 2354 y Fv(0)1989 2342 y Fz(;)14 b(t)2056 2354 y Fv(1)2093 2342 y FA(])31 b Fy(!)g Fz(X)2330 2354 y Fw(d)p Fu(\000)p Fw(d)2452 2362 y Ft(1)2519 2342 y FA(and)h(b)r(oth)h(parts)e(of)38 b(\(2.2\))515 2442 y(coincide)24 b(as)g(curv)n(es)f(in)i Fz(X)1343 2454 y Fw(d)p Fu(\000)p Fw(d)1465 2462 y Ft(1)1500 2442 y FA(.)36 b(A)25 b(solution)f Fz(x)h FA(is)f(called)g Fp(smo)l(oth)32 b FA(if)25 b(it)g(de\014nes)f(a)g(smo)r(oth)515 2541 y(curv)n(e)i(in)i(eac)n(h)f(space)g Fz(X)1311 2553 y Fw(s)1346 2541 y FA(.)639 2641 y(If)i(a)f(solution)g Fz(x)p FA(\()p Fz(t)p FA(\),)h Fz(t)24 b Fy(\025)g Fz(t)1474 2653 y Fv(0)1511 2641 y FA(,)29 b(of)35 b(\(2.2\))28 b(suc)n(h)g(that)g Fz(x)p FA(\()p Fz(t)2341 2653 y Fv(0)2379 2641 y FA(\))d(=)f Fz(x)2572 2653 y Fv(0)2638 2641 y FA(exists)j(and)i(is)f(unique,)515 2741 y(w)n(e)g(write)h Fz(x)p FA(\()p Fz(t)961 2753 y Fv(1)999 2741 y FA(\))c(=)g Fz(S)1202 2704 y Fw(t)1227 2712 y Ft(1)1197 2761 y Fw(t)1222 2769 y Ft(0)1263 2741 y Fz(x)1310 2753 y Fv(0)1348 2741 y FA(,)k(or)f Fz(x)p FA(\()p Fz(t)1612 2753 y Fv(1)1650 2741 y FA(\))e(=)f Fz(S)1854 2711 y Fw(t)1879 2719 y Ft(1)1911 2711 y Fu(\000)p Fw(t)1988 2719 y Ft(0)2024 2741 y Fz(x)2071 2753 y Fv(0)2138 2741 y FA(if)k(the)h(equation)e(is)h (autonomous)e(\(w)n(e)515 2850 y(do)g(not)h(assume)f(that)h Fz(t)1275 2862 y Fv(1)1335 2850 y Fy(\025)23 b Fz(t)1453 2862 y Fv(0)1490 2850 y FA(\).)38 b(The)28 b(op)r(erators)d Fz(S)2177 2813 y Fw(t)2202 2821 y Ft(1)2172 2871 y Fw(t)2197 2879 y Ft(0)2266 2850 y FA(and)j Fz(S)2484 2820 y Fw(t)2541 2850 y FA(are)e(called)i Fp(\015ow-maps)35 b FA(of)515 2960 y(the)28 b(equation.)37 b(Clearly)-7 b(,)27 b Fz(S)1391 2923 y Fw(t)1416 2931 y Ft(1)1386 2980 y Fw(t)1411 2988 y Ft(0)1480 2960 y FA(equals)g(\()p Fz(S)1820 2923 y Fw(t)1845 2931 y Ft(0)1815 2980 y Fw(t)1840 2988 y Ft(1)1882 2960 y FA(\))1914 2930 y Fu(\000)p Fv(1)2031 2960 y FA(on)g(a)h(join)n (t)g(domain)f(of)h(de\014nition)g(of)g(the)515 3060 y(t)n(w)n(o)f(op)r (erators.)639 3159 y(A)32 b(nonlinear)e(PDE)h(is)g(called)g Fp(str)l(ongly)i(nonline)l(ar)41 b FA(if)32 b(its)f(nonlinear)f(part)h (con)n(tains)515 3259 y(as)g(man)n(y)g(deriv)-5 b(ativ)n(es)30 b(as)h(the)h(linear)f(part.)49 b(Strongly)31 b(nonlinear)f(Hamiltonian) i(PDEs)515 3359 y(ma)n(y)21 b(p)r(ossess)f(rather)h(unpleasan)n(t)g (prop)r(erties.)34 b(In)22 b(particular,)g(for)f(some)g(of)h(them,)h (ev)n(ery)515 3458 y(non-zero)e(solution)h(dev)n(elops)g(a)h (singularit)n(y)e(in)i(\014nite)h(time,)g(see)f(an)f(example)h(in)g (Section)515 3558 y(1.4)j(of)i([Kuk00)o(].)639 3657 y(W)-7 b(e)27 b(shall)e(call)h(an)f(HPDE)h Fp(quasiline)l(ar)36 b FA(if)27 b(its)f(nonlinear)f(part)h(con)n(tains)f(less)g(deriv)-5 b(a-)515 3757 y(tiv)n(es)34 b(then)i(the)f(linear)f(one.)59 b(A)36 b(quasilinear)d(equation)i(can)f(b)r(e)i(written)f(in)g(the)g (form)515 3857 y(\(2.2\))27 b(with)1428 3956 y Fz(h)p FA(\()p Fz(x;)14 b(t)p FA(\))24 b(=)1775 3924 y Fv(1)p 1775 3938 34 4 v 1775 3985 a(2)1819 3956 y Fy(h)p Fz(Ax;)14 b(x)p Fy(i)20 b FA(+)e Fz(h)2227 3968 y Fv(0)2264 3956 y FA(\()p Fz(x;)c(t)p FA(\))p Fz(;)743 b FA(\(2.3\))515 4090 y(where)30 b Fz(A)h FA(is)g(a)f(linear)g(op)r(erator)f(whic)n(h)h (de\014nes)h(a)f(selfadjoin)n(t)h(morphism)f(of)h(the)g(scale)515 4189 y(\(so)c Fy(r)p Fz(h)p FA(\()p Fz(x;)14 b(t)p FA(\))24 b(=)f Fz(Ax)c FA(+)f Fy(r)p Fz(h)1384 4201 y Fv(0)1421 4189 y FA(\()p Fz(x;)c(t)p FA(\)\))29 b(and)e(ord)13 b Fy(r)p Fz(h)2072 4201 y Fv(0)2133 4189 y Fz(<)22 b FA(ord)13 b Fz(A)p FA(.)639 4289 y(The)22 b(class)e(of)i(Hamiltonian)f (PDEs)g(con)n(tains)f(man)n(y)h(imp)r(ortan)n(t)g(equations)g(of)g (math-)515 4389 y(ematical)36 b(ph)n(ysics,)j(some)e(of)g(them)h(are)e (discussed)h(b)r(elo)n(w.)66 b(The)37 b(\014rst)g(di\016cult)n(y)g(one) 515 4488 y(comes)i(across)g(when)h(studies)g(this)h(class)e(is)h (absence)g(of)g(a)g(general)e(theorem)i(whic)n(h)515 4588 y(w)n(ould)c(guaran)n(tee)e(that)j(\(lo)r(cally)f(in)h(time\))g (an)f(equation)g(has)f(a)h(unique)h(solution.)3342 4558 y Fv(2)515 4687 y FA(Suc)n(h)27 b(a)f(theorem)h(exists)f(for)h (semilinear)f(equations,)g(where)h(an)f(equation)h(\(2.2\))f(will)i(b)r (e)515 4787 y(called)i Fp(semiline)l(ar)41 b FA(if)31 b(its)g(Hamiltonian)g(has)f(the)h(form)f(\(2.3\))h(and)f(ord)13 b Fz(J)8 b Fy(r)p Fz(h)3009 4799 y Fv(0)3075 4787 y Fy(\024)28 b FA(0)i(\(see)515 4887 y([P)n(az83)m(])e(and)g(Section)f(1.4)g(of)g ([Kuk00)o(]\).)p 515 4929 1146 4 v 607 4983 a Fm(2)642 5006 y Fl(Still,)22 b(see)i([Kat75)q(])f(for)g(a)h(theory)h(whic)n(h)e (applies)h(to)g(some)f(classes)h(of)f(quasilinear)g(HPDEs.)1926 5255 y FA(6)p eop PStoPSsaved restore %%Page: (6,7) 4 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 7 6 bop 515 523 a Fp(Example)30 b FA(2.7)22 b Fp(\(e)l(quations)h(of)h (the)f(Kortewe)l(g{de)h(V)-6 b(ries)23 b(typ)l(e\).)35 b FA(Let)21 b(us)f(tak)n(e)f(for)h Fy(f)p Fz(X)3167 535 y Fw(s)3202 523 y Fy(g)g FA(the)515 623 y(scale)26 b(of)g(zero)g (mean-v)-5 b(alue)25 b(Sob)r(olev)h(spaces)g Fz(H)2053 593 y Fw(s)2088 623 y FA(\()p Fz(S)2176 593 y Fv(1)2213 623 y FA(\))2245 635 y Fv(0)2310 623 y FA(as)g(in)g(Example)g(2.1)g (and)g(c)n(ho)r(ose)515 736 y Fz(J)41 b FA(=)32 b Fz(@)5 b(=@)g(x)p FA(,)34 b(so)f Fz(d)1094 748 y Fw(J)1174 736 y FA(=)f(1.)54 b(F)-7 b(or)33 b(a)g(Hamiltonian)g Fz(h)h FA(w)n(e)f(tak)n(e)f Fz(h)p FA(\()p Fz(u)p FA(\))h(=)2789 669 y Fq(R)2844 689 y Fv(2)p Fw(\031)2828 765 y Fv(0)2922 736 y FA(\()p Fy(\000)3029 703 y Fv(1)p 3029 717 34 4 v 3029 764 a(8)3072 736 y Fz(u)3120 705 y Fu(0)3143 736 y FA(\()p Fz(x)p FA(\))3254 705 y Fv(2)3314 736 y FA(+)515 843 y Fz(f)9 b FA(\()p Fz(u)p FA(\)\))14 b Fz(dx)27 b FA(with)g(some)f(analytic)g(function)h Fz(f)9 b FA(\()p Fz(u)p FA(\).)36 b(Then)27 b Fy(r)p Fz(h)p FA(\()p Fz(u)p FA(\))c(=)2661 811 y Fv(1)p 2661 825 V 2661 872 a(4)2704 843 y Fz(u)2752 813 y Fu(00)2811 843 y FA(+)16 b Fz(f)2942 813 y Fu(0)2964 843 y FA(\()p Fz(u)p FA(\))27 b(and)g(the)515 943 y(equation)g(tak)n(es)f(the)i(form)1481 1164 y(_)-38 b Fz(u)p FA(\()p Fz(t;)14 b(x)p FA(\))24 b(=)1813 1108 y(1)p 1813 1145 42 4 v 1813 1221 a(4)1879 1164 y Fz(u)1927 1130 y Fu(000)2006 1164 y FA(+)2123 1108 y Fz(@)p 2099 1145 97 4 v 2099 1221 a(@)5 b(x)2219 1164 y(f)2269 1130 y Fu(0)2292 1164 y FA(\()p Fz(u)p FA(\))p Fz(:)515 1383 y FA(F)-7 b(or)31 b Fz(f)9 b FA(\()p Fz(u)p FA(\))31 b(=)966 1350 y Fv(1)p 966 1364 34 4 v 966 1411 a(4)1009 1383 y Fz(u)1057 1353 y Fv(3)1126 1383 y FA(w)n(e)h(get)g(the)h (classical)d(Kortew)n(eg{de)g(V)-7 b(ries)32 b(\(KdV\))h(equation.)50 b(The)515 1482 y(map)26 b Fz(V)746 1494 y Fw(h)817 1482 y FA(de\014nes)h(an)f(analytic)g(morphism)h(of)f(order)g(3)g(of)h(the)g (scale)f Fy(f)p Fz(X)2829 1494 y Fw(s)2864 1482 y Fy(g)p FA(,)h(for)f Fz(s)d(>)f FA(1)p Fz(=)p FA(2.)515 1582 y(The)j(equation)g(has)f(the)i(form)f(\(2.2\))o(,)h(\(2.3\),)g(where)e (ord)13 b Fz(J)8 b(A)24 b FA(=)e(3)j(and)g(ord)13 b Fz(J)8 b Fy(r)p Fz(h)3068 1594 y Fv(0)3128 1582 y FA(=)23 b(1.)36 b(It)515 1682 y(is)27 b(quasilinear,)f(but)j(not)e(semilinear.)p 3318 1682 4 57 v 3322 1629 50 4 v 3322 1682 V 3372 1682 4 57 v 515 1815 a Fp(Example)45 b FA(2.8)37 b Fp(\(NLS)h({)h(nonline)l (ar)f(Schr\177)-42 b(odinger)40 b(e)l(quation\).)48 b FA(Let)37 b Fz(X)2806 1827 y Fw(s)2880 1815 y FA(=)h Fz(H)3059 1784 y Fw(s)3094 1815 y FA(\()p Fx(T)3182 1784 y Fw(n)3227 1815 y FA(;)14 b Fx(C)h FA(\),)515 1914 y(where)25 b(this)h(Sob)r(olev)g(space)f(is)h(treated)f(as)h(a)f(real)g(Hilb)r (ert)i(space,)e(and)h(the)g(basic)g(scalar)515 2014 y(pro)r(duct)g(of)g (the)g(scale)f(is)h Fy(h)p Fz(u;)14 b(v)s Fy(i)23 b FA(=)g(Re)1753 1947 y Fq(R)1822 2014 y Fz(u)p 1870 1968 44 4 v(v)17 b(dx)p FA(.)37 b(F)-7 b(or)25 b Fz(J)34 b FA(w)n(e)26 b(tak)n(e)f(the)i(op)r(erator)d Fz(J)8 b(u)p FA(\()p Fz(x)p FA(\))23 b(=)515 2113 y Fz(iu)p FA(\()p Fz(x)p FA(\),)28 b(so)f(ord)14 b Fz(J)30 b FA(=)22 b(0)27 b(and)h(\()p Fy(f)p Fz(X)1528 2125 y Fw(s)1563 2113 y Fy(g)p Fz(;)p 1642 2047 55 4 v 14 w(J)7 b(du)19 b Fy(^)f Fz(du)p FA(\))28 b(is)g(a)f(Darb)r(oux)g(scale.)36 b(W)-7 b(e)28 b(c)n(ho)r(ose)1082 2338 y Fz(h)p FA(\()p Fz(u)p FA(\))23 b(=)1363 2282 y(1)p 1363 2319 42 4 v 1363 2395 a(2)1428 2225 y Fq(Z)1474 2414 y Fo(T)1519 2398 y Fn(n)1572 2271 y Fq(\000)1624 2338 y Fy(jr)p Fz(u)p Fy(j)1787 2304 y Fv(2)1842 2338 y FA(+)18 b Fz(V)h FA(\()p Fz(x)p FA(\))p Fy(j)p Fz(u)p Fy(j)2197 2304 y Fv(2)2253 2338 y FA(+)f Fz(g)s FA(\()p Fz(x;)c(u;)19 b FA(\026)-47 b Fz(u)p FA(\))2660 2271 y Fq(\001)2698 2338 y Fz(dx;)515 2568 y FA(where)33 b Fz(V)52 b FA(is)34 b(a)f(smo)r(oth)g(real)f(function)j(and)e Fz(g)s FA(\()p Fz(x;)14 b(u;)g(v)s FA(\))34 b(is)f(a)g(smo)r(oth)g (function,)j(real)d(if)515 2667 y Fz(v)26 b FA(=)i(\026)-47 b Fz(u)o FA(.)37 b(Then)28 b Fy(r)p Fz(h)p FA(\()p Fz(u)p FA(\))23 b(=)g Fy(\000)p FA(\001)p Fz(u)17 b FA(+)h Fz(V)h FA(\()p Fz(x)p FA(\))p Fz(u)g FA(+)1973 2635 y Fw(@)p 1953 2649 79 4 v 1953 2696 a(@)8 b Fv(\026)-37 b Fw(u)2042 2667 y Fz(g)30 b FA(and)d(\(2.2\))h(tak)n(es)e(the)i(form)773 2901 y(_)-37 b Fz(u)22 b FA(=)h Fz(i)946 2834 y Fq(\000)1002 2901 y Fy(\000)18 b FA(\001)p Fz(u)g FA(+)g Fz(V)h FA(\()p Fz(x)p FA(\))p Fz(u)g FA(+)1665 2845 y Fz(@)p 1641 2882 97 4 v 1641 2958 a(@)10 b FA(\026)-47 b Fz(u)1761 2901 y(g)s FA(\()p Fz(x;)14 b(u;)19 b FA(\026)-47 b Fz(u)o FA(\))2084 2834 y Fq(\001)2123 2901 y Fz(;)97 b(u)22 b FA(=)h Fz(u)p FA(\()p Fz(t;)14 b(x)p FA(\))p Fz(;)42 b(x)23 b Fy(2)h Fx(T)2897 2867 y Fw(n)2941 2901 y Fz(:)244 b FA(\(2.4\))515 3112 y(This)35 b(is)g(a)f(semilinear)h(Hamiltonian)f (equation)h(in)g(an)n(y)f(space)h Fz(X)2678 3124 y Fw(d)2713 3132 y Ft(0)2749 3112 y FA(,)i Fz(d)2852 3124 y Fv(0)2925 3112 y Fz(>)e(n=)p FA(2,)h(with)515 3211 y(ord)14 b Fz(A)22 b FA(=)h(2)k(and)g(ord)14 b Fy(r)p Fz(h)1304 3223 y Fv(0)1364 3211 y FA(=)22 b(0.)639 3311 y(Equation)27 b(\(2.4\))g(with)h Fz(n)23 b FA(=)g(1)p Fz(;)k(V)19 b FA(\()p Fz(x)p FA(\))24 b(=)14 b(const)27 b(and)h Fz(g)d FA(=)e Fz(\015)5 b Fy(j)p Fz(u)p Fy(j)2593 3281 y Fv(4)2630 3311 y Fz(;)41 b(\015)28 b Fy(6)p FA(=)23 b(0,)k(is)g(called)h(the)515 3411 y Fp(Zakhar)l(ov{Shab)l(at)38 b(e)l(quation)p FA(.)54 b(The)34 b(equation)e(with)i Fz(\015)k(>)32 b FA(0)h(is)g(called)g Fp(defo)l(cusing)42 b FA(and)515 3510 y(with)28 b Fz(\015)f(<)c FA(0)k({)h Fp(fo)l(cusing)7 b FA(.)p 3318 3510 4 57 v 3322 3457 50 4 v 3322 3510 V 3372 3510 4 57 v 515 3643 a Fp(Example)42 b FA(2.9)34 b Fp(\(1D)h(NLS)f(with)i(Dirichlet)g(b)l (oundary)g(c)l(onditions\).)46 b FA(Let)33 b(us)h(c)n(ho)r(ose)e(for) 515 3743 y Fz(X)584 3755 y Fw(s)647 3743 y FA(the)c(space)f Fz(H)1088 3713 y Fw(s)1081 3763 y Fv(0)1123 3743 y FA(\(0)p Fz(;)14 b(\031)s FA(;)g Fx(C)h FA(\))34 b(\(see)27 b(Example)g(2.2\),)g Fz(J)8 b(u)p FA(\()p Fz(x)p FA(\))24 b(=)e Fz(iu)p FA(\()p Fz(x)p FA(\))28 b(and)1097 3972 y Fz(h)p FA(\()p Fz(u)p FA(\))23 b(=)1378 3916 y(1)p 1378 3953 42 4 v 1378 4029 a(2)1443 3859 y Fq(Z)1526 3880 y Fw(\031)1489 4048 y Fv(0)1585 3905 y Fq(\000)1623 3972 y Fy(j)p Fz(u)1694 3984 y Fw(x)1735 3972 y Fy(j)1758 3938 y Fv(2)1814 3972 y FA(+)18 b Fz(V)h FA(\()p Fz(x)p FA(\))p Fy(j)p Fz(u)p Fy(j)2169 3938 y Fv(2)2225 3972 y FA(+)g Fz(g)s FA(\()p Fz(x;)14 b Fy(j)p Fz(u)p Fy(j)2562 3938 y Fv(2)2599 3972 y FA(\))2631 3905 y Fq(\001)2683 3972 y Fz(dx;)515 4201 y FA(where)42 b Fz(g)j FA(is)d(smo)r(oth)g(and)g(2)p Fz(\031)s FA(-p)r(erio)r(dic)g(in)h Fz(x)p FA(.)82 b(No)n(w)42 b Fy(r)p Fz(h)p FA(\()p Fz(u)p FA(\))48 b(=)f Fy(\000)p Fz(u)2860 4213 y Fw(xx)2967 4201 y FA(+)28 b Fz(V)19 b FA(\()p Fz(x)p FA(\))p Fz(u)28 b FA(+)515 4301 y Fz(f)9 b FA(\()p Fz(x;)14 b Fy(j)p Fz(u)p Fy(j)775 4271 y Fv(2)812 4301 y FA(\))p Fz(u)p FA(,)27 b(where)g Fz(f)32 b FA(=)1391 4264 y Fw(@)t(g)p 1353 4282 151 4 v 1353 4330 a(@)t Fu(j)p Fw(u)p Fu(j)1471 4313 y Ft(2)1513 4301 y FA(,)c(and)f(\(2.2\))g(b)r (ecomes)976 4507 y(_)-38 b Fz(u)23 b FA(=)g Fz(i)p FA(\()p Fy(\000)p Fz(u)1294 4519 y Fw(xx)1390 4507 y FA(+)18 b Fz(V)h FA(\()p Fz(x)p FA(\))p Fz(u)g FA(+)f Fz(f)9 b FA(\()p Fz(x;)14 b Fy(j)p Fz(u)p Fy(j)2061 4473 y Fv(2)2098 4507 y FA(\))p Fz(u)p FA(\))p Fz(;)97 b(u)p FA(\(0\))23 b(=)g Fz(u)p FA(\()p Fz(\031)s FA(\))g(=)g(0)p Fz(:)275 b FA(\(2.5\))515 4690 y(F)-7 b(or)39 b Fz(s)44 b FA(=)g(1)c(and)g(2)g (the)g(nonlinear)g(term)g(de\014nes)g(a)g(smo)r(oth)g(map)g Fz(X)2889 4702 y Fw(s)2968 4690 y Fy(!)45 b Fz(X)3165 4702 y Fw(s)3240 4690 y FA(\(see)515 4790 y(Example)32 b(2.5\),)h(so)f(in)h(these)g(spaces)f(this)h(is)g(a)f(semilinear)g (equation)g(with)h(ord)14 b Fz(A)31 b FA(=)g(2)515 4889 y(and)d(ord)14 b Fy(r)p Fz(h)929 4901 y Fv(0)989 4889 y FA(=)23 b(0.)38 b(If)29 b(in)f(addition)g Fz(f)37 b FA(is)28 b(ev)n(en)g(in)g Fz(x)p FA(,)h(then)f(the)h(nonlinear)e(term)h (de\014nes)515 4989 y(a)f(smo)r(oth)g(map)h(for)f(ev)n(ery)f Fz(s)d Fy(\025)g FA(1.)36 b(This)28 b(map)f(is)h(analytic)f(if)h Fz(f)36 b FA(is.)p 3318 4989 4 57 v 3322 4936 50 4 v 3322 4989 V 3372 4989 4 57 v 1926 5255 a(7)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 8 7 bop 515 523 a FC(3)134 b(Basic)45 b(theorems)h(on)f(Hamiltonian)i (systems)515 705 y FA(Basic)32 b(theorems)h(from)f(the)i(classical)e (Hamiltonian)h(formalism)f(\(see)h([Arn89)o(,)h(HZ94)o(]\))515 805 y(remain)23 b(true)h(for)g(Hamiltonian)g(equations)f(\(2.2\))h(in)h (Hilb)r(ert)f(scales,)g(pro)n(vided)f(that)i(the)515 904 y(theorems)h(are)g(prop)r(erly)g(form)n(ulated.)36 b(In)28 b(this)f(section)g(w)n(e)g(presen)n(t)f(three)h(corresp)r(ond-) 515 1004 y(ing)g(results.)36 b(Their)28 b(pro)r(ofs)e(can)i(b)r(e)g (found)g(in)f([Kuk93)o(,)h(Kuk00)n(].)639 1103 y(Let)34 b(\()p Fy(f)p Fz(X)937 1115 y Fw(s)972 1103 y Fy(g)p Fz(;)14 b(\013)1104 1115 y Fv(2)1173 1103 y FA(=)p 1269 1037 55 4 v 31 w Fz(J)22 b(dx)h Fy(^)g Fz(dx)p FA(\))34 b(and)f(\()p Fy(f)p Fz(Y)1973 1115 y Fw(s)2008 1103 y Fy(g)p Fz(;)14 b(\014)2134 1115 y Fv(2)2203 1103 y FA(=)p 2300 1037 65 4 v 32 w(\007)f Fz(dy)25 b Fy(^)e Fz(dy)s FA(\))33 b(b)r(e)h(t)n(w)n(o)e(symplectic)515 1203 y(scales)f(and)i (\(for)f(simplicit)n(y\))h(ord)13 b Fz(J)39 b FA(=)31 b(ord)13 b(\007)31 b(=)g Fz(d)2182 1215 y Fw(J)2260 1203 y Fy(\025)g FA(0.)51 b(Let)33 b(\010)9 b(:)29 b Fz(Q)i Fy(!)g Fz(O)36 b FA(b)r(e)d(a)f Fz(C)3314 1173 y Fv(1)3351 1203 y FA(-)515 1303 y(smo)r(oth)e(symplectic)h(map,)g(where)f Fz(Q)g FA(and)g Fz(O)k FA(are)29 b(domains)h(in)h Fz(Y)2654 1315 y Fw(d)2723 1303 y FA(and)g Fz(X)2957 1315 y Fw(d)2995 1303 y FA(,)h Fz(d)c Fy(\025)f FA(0.)46 b(If)515 1402 y Fz(d)558 1414 y Fw(J)627 1402 y Fz(>)23 b FA(0,)k(w)n(e)g(ha)n(v)n(e) g(to)g(assume)g(that)512 1585 y(\(H1\))42 b(for)27 b(an)n(y)f Fy(j)p Fz(s)p Fy(j)d(\024)g Fz(d)k FA(linearised)g(maps)f(\010)1917 1597 y Fu(\003)1956 1585 y FA(\()p Fz(y)s FA(\),)h Fz(y)f Fy(2)d Fz(Q)p FA(,)k(de\014ne)h(linear)e(maps)h Fz(Y)3110 1597 y Fw(s)3169 1585 y Fy(!)c Fz(X)3344 1597 y Fw(s)722 1685 y FA(whic)n(h)28 b(con)n(tin)n(uously)e(dep)r(end)j(on)e Fz(y)s FA(.)639 1867 y(The)e(\014rst)g(theorem)g(states)g(that)g (symplectic)g(maps)g(transform)f(Hamiltonian)h(equa-)515 1967 y(tions)i(to)h(Hamiltonian:)515 2133 y FB(Theorem)38 b(3.1.)44 b Fp(L)l(et)36 b FA(\010)9 b(:)30 b Fz(Q)k Fy(!)g Fz(O)39 b Fp(b)l(e)d(a)g(symple)l(ctic)h(map)g(as)f(ab)l(ove)43 b FA(\()p Fp(so)f FA(\(H1\))37 b Fp(holds)515 2232 y(if)e Fz(d)643 2244 y Fw(J)722 2232 y Fz(>)d FA(0\))p Fp(.)53 b(L)l(et)35 b(us)f(assume)g(that)h(the)g(ve)l(ctor)g(\014eld)g Fz(V)2327 2244 y Fw(h)2405 2232 y Fp(of)h(e)l(quation)41 b FA(\(2.2\))35 b Fp(de\014nes)g(a)515 2332 y Fz(C)580 2302 y Fv(1)617 2332 y Fp(-smo)l(oth)30 b(map)g Fz(V)1159 2344 y Fw(h)1212 2332 y FA(:)e Fz(O)21 b Fy(\002)c Fx(R)29 b Fy(!)23 b Fz(X)1687 2344 y Fw(d)p Fu(\000)p Fw(d)1809 2352 y Ft(1)1874 2332 y Fp(of)31 b(or)l(der)f Fz(d)2233 2344 y Fv(1)2294 2332 y Fy(\024)22 b FA(2)p Fz(d)30 b Fp(and)g(that)g(this)g(ve)l(ctor)f(\014eld)515 2432 y(is)e(tangent)f (to)h(the)g(map)h FA(\010)f(\()p Fp(i.e.,)j(for)e(every)g Fz(y)d Fy(2)f Fz(Q)i Fp(and)i(every)g Fz(t)e Fp(the)i(ve)l(ctor)f Fz(V)3036 2444 y Fw(h)3079 2432 y FA(\(\010\()p Fz(y)s FA(\))p Fz(;)14 b(t)p FA(\))515 2531 y Fp(b)l(elong)31 b(to)h(the)f(r)l(ange)g(of)h(the)g(line)l(arise)l(d)g(map)g FA(\010)2088 2543 y Fu(\003)2126 2531 y FA(\()p Fz(y)s FA(\)\))p Fp(.)44 b(Then)31 b FA(\010)g Fp(tr)l(ansforms)h(solutions) 515 2631 y(of)f(the)f(Hamiltonian)h(e)l(quation)46 b FA(_)-38 b Fz(y)26 b FA(=)e(\007)p Fy(r)1853 2643 y Fw(y)1893 2631 y Fz(H)7 b FA(\()p Fz(y)s(;)14 b(t)p FA(\))p Fp(,)30 b(wher)l(e)h Fz(H)g FA(=)23 b Fz(h)c Fy(\016)f FA(\010)p Fz(;)30 b Fp(to)h(solutions)f(of)515 2731 y FA(\(2.2\))515 2897 y FB(Corollary)43 b(3.2.)i Fp(If)38 b(under)f(the)h(assumptions)f (of)i(The)l(or)l(em)45 b FA(3.1)36 b Fy(f)p Fz(X)2828 2909 y Fw(s)2863 2897 y Fy(g)h FA(=)f Fy(f)p Fz(Y)3133 2909 y Fw(s)3168 2897 y Fy(g)h Fp(and)515 2996 y FA(\010)575 2966 y Fu(\003)613 2996 y Fz(\013)666 3008 y Fv(2)726 2996 y FA(=)23 b Fz(\013)867 3008 y Fv(2)904 2996 y Fp(,)30 b(then)g FA(\010)g Fp(pr)l(eserves)g(the)g(class)h(of)f(solutions)g (for)40 b FA(\(2.2\))o Fp(.)639 3162 y FA(F)-7 b(or)29 b(Hamiltonian)h(PDEs)f(\(and)h(for)f(Hamiltonian)g(equations)g (\(2.2\)\))h(Theorem)f(2.1)515 3262 y(pla)n(ys)22 b(the)i(same)f(role)f (as)h(its)g(classical)f(\014nite-dimensional)h(coun)n(terpart)f(pla)n (ys)h(for)f(usual)515 3362 y(Hamiltonian)31 b(equations:)43 b(it)31 b(is)g(used)g(to)g(transform)f(an)h(equation)g(to)g(a)f(normal) g(form,)515 3461 y(usually)d(in)h(the)g(vicinit)n(y)f(of)h(an)f(in)n(v) -5 b(arian)n(t)27 b(set)g(\(e.g.,)h(of)f(an)h(equilibrium\).)639 3561 y(T)-7 b(o)23 b(apply)h(Theorem)e(3.1)h(one)g(needs)g(regular)f(w) n(a)n(ys)g(to)i(construct)f(symplectic)g(trans-)515 3660 y(formations.)35 b(F)-7 b(or)26 b(classical)f(\014nite-dimensional)h (systems)g(symplectic)g(transformations)515 3760 y(usually)k(are)f (obtained)h(either)g(via)g(generating)e(functions,)k(or)d(as)h(Lie)g (transformations)515 3860 y(\(i.e.,)h(as)f(\015o)n(w-maps)f(of)h (additional)g(Hamiltonians\),)h(see)f([Arn89)o(,)h(HZ94)o(,)f(Gia72)o (].)46 b(F)-7 b(or)515 3959 y(in\014nite)27 b(dimensional)f(symplectic) h(spaces)f(generating)f(functions)i(pla)n(y)f(negligible)g(role,)515 4059 y(while)c(the)h(Lie)f(transformations)f(remain)g(an)h(imp)r(ortan) n(t)g(to)r(ol.)35 b(An)23 b(easy)e(but)i(imp)r(ortan)n(t)515 4159 y(corresp)r(onding)i(result)j(is)f(stated)h(in)g(the)g(theorem)f (b)r(elo)n(w.)639 4258 y(Let)33 b(\()p Fy(f)p Fz(X)936 4270 y Fw(s)971 4258 y Fy(g)p Fz(;)14 b(\013)1103 4270 y Fv(2)1140 4258 y FA(\))33 b(b)r(e)g(a)g(symplectic)f(Hilb)r(ert)i (scale)e(as)g(ab)r(o)n(v)n(e)f(and)i Fz(O)i FA(b)r(e)e(a)f(domain)515 4358 y(in)c Fz(X)681 4370 y Fw(d)719 4358 y FA(.)515 4524 y FB(Theorem)39 b(3.3.)45 b Fp(L)l(et)37 b Fz(f)46 b Fp(b)l(e)37 b(a)g Fz(C)1615 4494 y Fv(1)1653 4524 y Fp(-smo)l(oth)g(function)g(on)h Fz(O)26 b Fy(\002)d Fx(R)44 b Fp(such)37 b(that)g(the)g(map)515 4624 y Fz(V)563 4636 y Fw(f)615 4624 y FA(:)c Fz(O)e Fy(\002)d Fx(R)53 b Fy(!)48 b Fz(X)1165 4636 y Fw(d)1246 4624 y Fp(is)c(Lipschitz)h(in)e FA(\()p Fz(x;)14 b(t)p FA(\))44 b Fp(and)g Fz(C)2291 4593 y Fv(1)2328 4624 y Fp(-smo)l(oth)g(in)f Fz(x)p Fp(.)80 b(L)l(et)42 b Fz(O)3140 4636 y Fv(1)3221 4624 y Fp(b)l(e)i(a)515 4723 y(sub)l(domain)37 b(of)h Fz(O)r Fp(.)60 b(Then)37 b(the)g(\015ow-maps)g Fz(S)2009 4693 y Fw(\034)2004 4744 y(t)2060 4723 y FA(:)30 b(\()p Fz(O)2208 4735 y Fv(1)2246 4723 y Fz(;)14 b(\013)2336 4735 y Fv(2)2373 4723 y FA(\))36 b Fy(!)f FA(\()p Fz(O)r(;)14 b(\013)2746 4735 y Fv(2)2784 4723 y FA(\))37 b Fp(ar)l(e)g(symple)l(cto-)515 4823 y(morphisms)j FA(\()p Fp(pr)l(ovide)l(d)35 b(that)d(they)h(map)g Fz(O)1909 4835 y Fv(1)1979 4823 y Fp(to)f Fz(O)r FA(\))p Fp(.)47 b(If)33 b(the)g(map)g Fz(V)2714 4835 y Fw(f)2789 4823 y Fp(is)g Fz(C)2946 4793 y Fw(k)2987 4823 y Fp(-smo)l(oth)g(or)515 4922 y(analytic,)e(then)f(the)g(\015ow-maps)g(ar)l(e)g Fz(C)1775 4892 y Fw(k)1816 4922 y Fp(-smo)l(oth)g(or)g(analytic)h(as)g (wel)t(l.)1926 5255 y FA(8)p eop PStoPSsaved restore %%Page: (8,9) 5 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 9 8 bop 639 523 a FA(The)30 b(assumption)e(that)i(the)f(map)g Fz(V)1812 535 y Fw(f)1885 523 y FA(is)g(Lipsc)n(hitz)g(can)g(b)r(e)h (replaced)e(b)n(y)h(the)g(m)n(uc)n(h)515 623 y(w)n(eak)n(er)38 b(assumption)i(that)h(for)f(a)g(solution)f Fz(x)p FA(\()p Fz(t)p FA(\))j(of)e(the)h(equation)54 b(_)-37 b Fz(x)45 b FA(=)e Fz(V)3041 635 y Fw(f)3085 623 y FA(\()p Fz(x)p FA(\),)i(the)515 732 y(linearised)23 b(equation)1235 710 y(_)1218 732 y Fz(\030)k FA(=)22 b Fz(V)1416 744 y Fw(f)7 b Fu(\003)1494 732 y FA(\()p Fz(x)p FA(\()p Fz(t)p FA(\)\))p Fz(\030)30 b FA(is)25 b(suc)n(h)f(that)h(its)g(\015o)n (w)f(maps)g(are)g(b)r(ounded)h(linear)515 832 y(transformations)g(of)j (the)g(space)f Fz(X)1642 844 y Fw(d)1680 832 y FA(.)37 b(See)28 b([Kuk00)n(].)639 931 y(Usually)c(Theorem)e(3.3)h(is)h (applied)g(in)g(the)g(situation)f(when)h Fy(j)p Fz(f)9 b Fy(j)23 b(\034)g FA(1,)h(or)f Fy(j)p Fz(t)11 b Fy(\000)g Fz(\034)e Fy(j)22 b(\034)h FA(1.)515 1031 y(In)33 b(these)f(cases)g (the)h(\015o)n(w-maps)e(are)h(closed)g(to)h(the)g(iden)n(tit)n(y)f(and) h(the)g(corresp)r(onding)515 1131 y(transformations)d(of)i(the)g(space) f(of)h Fz(C)1754 1100 y Fv(1)1792 1131 y FA(-smo)r(oth)f(functions)i (on)e Fz(O)r FA(,)j Fz(H)k Fy(7!)30 b Fz(H)e Fy(\016)21 b Fz(S)3157 1100 y Fw(\034)3152 1151 y(t)3198 1131 y FA(,)33 b(can)515 1230 y(b)r(e)d(written)g(as)g(Lie)g(series)e(\(cf.)j ([Gia72)o(]\).)44 b(In)31 b(particular,)e(the)h(follo)n(wing)f(simple)h (result)515 1330 y(holds:)515 1466 y FB(Theorem)i(3.4.)41 b Fp(Under)31 b(the)g(assumptions)g(of)h(The)l(or)l(em)38 b FA(3.3)p Fp(,)31 b(let)g Fz(H)38 b Fp(b)l(e)31 b(a)g Fz(C)3059 1436 y Fv(1)3097 1466 y Fp(-smo)l(oth)515 1565 y(function)e(on)h Fz(O)r Fp(.)40 b(Then)1191 1693 y Fz(d)p 1169 1730 89 4 v 1169 1806 a(d\034)1281 1749 y(H)7 b FA(\()p Fz(S)1445 1715 y Fw(\034)1440 1770 y(t)1487 1749 y FA(\()p Fz(x)p FA(\)\))24 b(=)f Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)p FA(\()p Fz(S)2072 1715 y Fw(\034)2067 1770 y(t)2112 1749 y FA(\()p Fz(x)p FA(\)\))p Fz(;)185 b(x)23 b Fy(2)g Fz(O)2674 1761 y Fv(1)2712 1749 y Fz(:)473 b FA(\(3.1\))639 1922 y(In)28 b(this)g(theorem)f Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)26 b FA(denotes)i(the)g Fp(Poisson)j(br)l (acket)36 b FA(of)27 b(the)h(t)n(w)n(o)f(functions:)1382 2067 y Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)p FA(\()p Fz(x)p FA(\))23 b(=)f Fy(h)p Fz(J)8 b Fy(r)p Fz(f)h FA(\()p Fz(x)p FA(\))p Fz(;)14 b Fy(r)p Fz(H)7 b FA(\()p Fz(x)p FA(\))p Fy(i)p Fz(:)515 2212 y FA(It)28 b(is)f(w)n(ell)h(de\014ned)f (since)h Fz(J)8 b Fy(r)p Fz(f)32 b FA(=)22 b Fz(V)1677 2224 y Fw(f)1744 2212 y Fy(2)h Fz(X)1891 2224 y Fw(d)1957 2212 y FA(b)n(y)28 b(assumptions.)639 2312 y(Theorem)j(3.3)g(and)g (form)n(ula)g(\(3.1\))g(mak)n(e)g(from)g(symplectic)h(\015o)n(w-maps)e Fz(S)3092 2282 y Fw(\034)3087 2333 y(t)3165 2312 y FA(a)h(to)r(ol)515 2412 y(whic)n(h)23 b(suits)g(w)n(ell)g(to)h(pro)n(v)n(e)d(KAM-theorems) h(for)h(Hamiltonian)g(PDEs,)h(see)f(the)h(sc)n(heme)515 2511 y(of)j(the)h(pro)r(of)f(of)h(Theorem)f(5.1)f(in)i(section)f(5.1)g (b)r(elo)n(w.)639 2611 y(An)22 b(immediate)g(consequence)e(of)h (Theorem)g(3.4)f(is)i(that)f(for)g(an)g(autonomous)f(Hamil-)515 2711 y(tonian)29 b(equation)43 b(_)-38 b Fz(x)26 b FA(=)g Fz(J)8 b Fy(r)p Fz(f)h FA(\()p Fz(x)p FA(\))30 b(suc)n(h)f(that)g(ord) 13 b Fz(J)8 b Fy(r)p Fz(f)35 b FA(=)25 b(0,)k(a)g Fz(C)2616 2680 y Fv(1)2654 2711 y FA(-smo)r(oth)f(function)i Fz(H)515 2810 y FA(is)d(an)h(in)n(tegral)e(of)h(motion)1392 2780 y Fv(3)1457 2810 y FA(if)h(and)f(only)g(if)i Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)21 b(\021)i FA(0.)639 2910 y(If)i Fz(d)762 2880 y Fu(0)808 2910 y FA(=)e(ord)13 b Fz(J)8 b Fy(r)p Fz(f)32 b(>)22 b FA(0)i(and)f Fz(O)j FA(=)d Fz(O)1776 2922 y Fw(d)1839 2910 y FA(b)r(elongs)g(to)g(a)h(system)f(of)h (compatible)g(domains)515 3010 y Fz(O)578 3022 y Fw(s)647 3010 y Fy(\032)33 b Fz(X)814 3022 y Fw(s)849 3010 y FA(,)j Fz(s)d Fy(2)h FA([)p Fz(d)1135 3022 y Fv(0)1172 3010 y Fz(;)14 b(d)p FA(],)36 b(where)d Fz(d)1623 3022 y Fv(0)1694 3010 y FA(=)g Fz(d)23 b Fy(\000)f Fz(d)1988 2979 y Fu(0)2011 3010 y FA(,)36 b(then)e Fz(H)41 b FA(suc)n(h)33 b(that)h Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)32 b(\021)h FA(0)h(is)f(an)515 3109 y(in)n(tegrable)26 b(of)i(motion)f(for)g(the)h(equation)42 b(_)-38 b Fz(x)24 b FA(=)e Fz(J)8 b Fy(r)p Fz(f)h FA(\()p Fz(x)p FA(\),)29 b(pro)n(vided)d(that)982 3254 y(ord)13 b Fz(J)8 b Fy(r)p Fz(f)32 b FA(=)23 b Fz(d)1443 3220 y Fu(0)1549 3254 y FA(and)97 b(ord)13 b Fy(r)p Fz(H)30 b FA(=)23 b Fz(d)2213 3266 y Fw(H)2359 3254 y FA(for)82 b Fz(s)23 b Fy(2)h FA([)p Fz(d)2748 3266 y Fv(0)2785 3254 y Fz(;)14 b(d)p FA(])p Fz(;)515 3399 y FA(where)35 b Fz(d)806 3369 y Fu(0)853 3399 y FA(+)23 b Fz(d)984 3411 y Fw(H)1083 3399 y Fy(\024)36 b FA(2)p Fz(d)p FA(.)60 b(Indeed,)38 b(since)d Fz(d)1909 3411 y Fv(0)1970 3399 y Fy(\000)23 b Fz(d)2101 3411 y Fw(H)2201 3399 y Fy(\025)35 b(\000)p Fz(d)2409 3411 y Fv(0)2446 3399 y FA(,)j(then)e Fz(H)42 b FA(is)35 b(a)g Fz(C)3048 3369 y Fv(1)3086 3399 y FA(-smo)r(oth)515 3499 y(function)e(on)g Fz(O)1029 3511 y Fw(d)1064 3519 y Ft(0)1100 3499 y FA(.)53 b(Since)33 b(an)n(y)f(solution)h Fz(x)p FA(\()p Fz(t)p FA(\))h(is)e(a)h Fz(C)2283 3469 y Fv(1)2320 3499 y FA(-smo)r(oth)g(curv)n(e)f(in)h Fz(O)3039 3511 y Fw(d)3074 3519 y Ft(0)3143 3499 y FA(b)n(y)g(the)515 3599 y(de\014nition)28 b(of)f(a)g(solution,)h(then)870 3726 y Fz(d)p 855 3763 74 4 v 855 3839 a(dt)952 3782 y(H)7 b FA(\()p Fz(x)p FA(\))24 b(=)e Fy(hr)p Fz(H)7 b FA(\()p Fz(x)p FA(\))p Fz(;)30 b FA(_)-39 b Fz(x)r Fy(i)23 b FA(=)g Fy(hr)p Fz(H)7 b FA(\()p Fz(x)p FA(\))p Fz(;)14 b(J)8 b Fy(r)p Fz(f)h FA(\()p Fz(x)p FA(\))p Fy(i)25 b FA(=)e Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)p FA(\()p Fz(x)p FA(\))23 b(=)f(0)p Fz(:)639 3955 y FA(In)28 b(particular,)e Fz(f)36 b FA(is)27 b(an)g(in)n(tegral)f(of)i(motion)f(for)g(the)g (equation)41 b(_)-37 b Fz(x)23 b FA(=)g Fz(J)8 b Fy(r)p Fz(f)h FA(\()p Fz(x)p FA(\))28 b(in)g Fz(O)3340 3967 y Fw(d)515 4055 y FA(if)j(w)n(e)f(ha)n(v)n(e)f(ord)13 b Fz(J)35 b FA(=)28 b Fz(d)1264 4067 y Fw(J)1341 4055 y FA(and)i(ord)13 b Fy(r)p Fz(f)36 b FA(=)28 b Fz(d)1921 4067 y Fw(f)1994 4055 y FA(for)i Fz(s)e FA(=)f Fz(d)k FA(and)f(for)g Fz(s)d Fy(2)h FA([)p Fz(d;)14 b(d)21 b Fy(\000)f Fz(d)3095 4067 y Fw(f)3158 4055 y Fy(\000)g Fz(d)3286 4067 y Fw(J)3333 4055 y FA(],)515 4155 y(where)25 b Fz(d)f Fy(\025)e Fz(d)950 4167 y Fw(f)1009 4155 y FA(+)15 b Fz(d)1132 4167 y Fw(J)1179 4155 y Fz(=)p FA(2.)35 b(That)26 b(is,)h(if)f(the)h(equation)e(is)h(b)r(eing)h(considered)e(in)h (su\016cien)n(tly)515 4254 y(smo)r(oth)h(spaces.)515 4372 y Fp(Example)37 b FA(3.5)p Fp(.)k FA(Let)28 b(us)g(consider)f(a)h (nonlinear)f(Sc)n(hr\177)-42 b(odinger)26 b(equation)h(\(2.5\))h(suc)n (h)g(that)515 4472 y Fz(g)s FA(\()p Fz(u;)p 675 4426 48 4 v 14 w(u)o FA(\))f(=)g Fz(g)913 4484 y Fv(0)950 4472 y FA(\()p Fy(j)p Fz(u)p Fy(j)1076 4442 y Fv(2)1113 4472 y FA(\),)k(and)f(tak)n(e)f Fz(H)7 b FA(\()p Fz(u)p FA(\))27 b(=)g Fy(k)p Fz(u)p Fy(k)1984 4442 y Fv(2)1984 4492 y(0)2047 4472 y FA(=)g Fy(j)p Fz(u)p Fy(j)2233 4442 y Fv(2)2233 4495 y Fw(L)2279 4503 y Ft(2)2315 4472 y FA(.)44 b(No)n(w)30 b Fz(d)2617 4442 y Fu(0)2667 4472 y FA(:=)d(ord)13 b Fz(J)8 b Fy(r)p Fz(f)36 b FA(=)27 b(2)j(for)515 4571 y Fz(s)23 b Fy(2)h FA(\()p Fz(n=)p FA(2)p Fz(;)14 b Fy(1)p FA(\),)27 b(and)h(ord)13 b Fy(r)p Fz(H)31 b FA(=)23 b(0.)37 b(Elemen)n(tary)26 b(calculations)h(sho)n(w)g (that)h Fy(f)p Fz(f)t(;)14 b(H)7 b Fy(g)22 b(\021)h FA(0.)515 4671 y(So)k Fz(L)687 4683 y Fv(2)724 4671 y FA(-norm)g(is)h(an)g(in)n (tegral)e(of)i(motion)g(for)g(solutions)f(of)34 b(\(2.5\))28 b(in)g Fz(X)2789 4683 y Fw(s)2852 4671 y FA(if)h Fz(s)23 b(>)h(n=)p FA(2)17 b(+)h(2.)515 4771 y(\(In)37 b(fact)f(this)h(result)f (remains)f(true)i(for)f(solutions)f(of)i(m)n(uc)n(h)f(lo)n(w)n(er)f (smo)r(othness,)i(see)515 4870 y([Bou93)n(])28 b(\).)p 3318 4870 4 57 v 3322 4818 50 4 v 3322 4870 V 3372 4870 4 57 v 515 4929 1146 4 v 607 4983 a Fm(3)642 5006 y Fl(That)c(is,)f Fk(H)5 b Fl(\()p Fk(x)p Fl(\()p Fk(t)p Fl(\)\))25 b(is)e(a)h (time-indep)r(enden)n(t)h(quan)n(tit)n(y)g(for)e(an)n(y)h(solution)g Fk(x)p Fl(\()p Fk(t)p Fl(\)\).)1926 5255 y FA(9)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 10 9 bop 515 523 a FC(4)134 b(Lax-in)l(tegrable)47 b(equations)515 721 y Fs(4.1)112 b(General)37 b(discussion)515 875 y FA(Let)d(us)h(tak)n(e)e(a)h(Hamiltonian)h(PDE)f(and)g(write)g(it)h(as)f (a)g(Hamiltonian)g(equation)g(in)g(a)515 974 y(suitable)27 b(symplectic)h(Hilb)r(ert)g(scale)f(\()p Fy(f)p Fz(X)1861 986 y Fw(s)1896 974 y Fy(g)p Fz(;)14 b(\013)2028 986 y Fv(2)2088 974 y FA(=)p 2175 908 55 4 v 22 w Fz(J)22 b(du)c Fy(^)h Fz(du)p FA(\):)1715 1135 y(_)-37 b Fz(u)22 b FA(=)h Fz(J)8 b Fy(r)p Fz(H)f FA(\()p Fz(u)p FA(\))p Fz(:)1015 b FA(\(4.1\))515 1296 y(This)26 b(equation)f(is)i(called)e (Lax-in)n(tegrable)f(if)j(there)f(exists)g(an)f(additional)h(Hilb)r (ert)h(scale)515 1395 y Fy(f)p Fz(Z)614 1407 y Fw(s)648 1395 y Fy(g)34 b FA(\(real)g(or)f(complex\),)j(and)e(\014nite)g(order)f (linear)g(morphisms)h Fy(L)2754 1407 y Fw(u)2832 1395 y FA(and)g Fy(A)3066 1407 y Fw(u)3144 1395 y FA(of)g(this)515 1495 y(scale)19 b(whic)n(h)h(dep)r(end)h(on)f(the)h(parameter)e Fz(u)k Fy(2)g Fz(X)2069 1507 y Fu(1)2139 1495 y FA(,)f(suc)n(h)e(that)h (a)e(curv)n(e)h Fz(u)p FA(\()p Fz(t)p FA(\))g(is)g(a)g(smo)r(oth)515 1595 y(solution)27 b(for)g(\(4.1\))g(if)h(and)g(only)f(if)1539 1742 y Fz(d)p 1524 1779 74 4 v 1524 1856 a(dt)1621 1799 y Fy(L)1678 1814 y Fw(u)p Fv(\()p Fw(t)p Fv(\))1822 1799 y FA(=)22 b([)p Fy(A)1998 1814 y Fw(u)p Fv(\()p Fw(t)p Fv(\))2119 1799 y Fz(;)14 b Fy(L)2213 1814 y Fw(u)p Fv(\()p Fw(t)p Fv(\))2334 1799 y FA(])p Fz(:)828 b FA(\(4.2\))515 1987 y(The)24 b(op)r(erators)f Fy(A)1113 1999 y Fw(u)1181 1987 y FA(and)h Fy(L)1396 1999 y Fw(u)1440 1987 y FA(,)h(treated)f(as)g (morphisms)g(of)g(the)h(scale)f Fy(f)p Fz(Z)2814 1999 y Fw(s)2848 1987 y Fy(g)p FA(,)h(are)f(assumed)515 2087 y(to)f(dep)r(end)g(smo)r(othly)g(on)g Fz(u)f Fy(2)i Fz(X)1580 2099 y Fw(d)1642 2087 y FA(where)e Fz(d)h FA(is)g(su\016cien)n(tly)g (large,)g(so)f(the)i(left-hand)f(side)515 2186 y(of)36 b(\(4.2\))29 b(is)g(w)n(ell)h(de\014ned)f(\(for)h(details)f(see)g ([Kuk00)o(]\).)43 b(The)29 b(pair)g(of)g(op)r(erators)f Fy(L)p FA(,)i Fy(A)g FA(is)515 2286 y(called)d(the)h Fp(L)l(ax)h(p)l(air)p FA(.)1252 2256 y Fv(4)639 2386 y FA(In)34 b(most)g(kno)n(wn)f(examples)g(of)g(Lax-in)n(tegrable)f (equations)g(relation)h(b)r(et)n(w)n(een)h(the)515 2485 y(scales)k Fy(f)p Fz(X)869 2497 y Fw(s)903 2485 y Fy(g)h FA(and)f Fy(f)p Fz(Z)1255 2497 y Fw(s)1290 2485 y Fy(g)g FA(is)h(the)g(follo)n(wing:)59 b(spaces)37 b Fz(X)2359 2497 y Fw(s)2434 2485 y FA(are)g(formed)i(b)n(y)f Fz(T)12 b FA(-p)r(erio)r(dic)515 2585 y(Sob)r(olev)32 b(v)n(ector-functions,)g (while)h Fy(A)g FA(and)f Fy(L)h FA(are)e(di\013eren)n(tial)h(or)g(in)n (tegro-di\013eren)n(tial)515 2685 y(op)r(erators)d(with)k Fz(u)p FA(-dep)r(enden)n(t)e(co)r(e\016cien)n(ts,)h(acting)f(in)h(a)g (scale)e Fy(f)p Fz(Z)2737 2697 y Fw(s)2772 2685 y Fy(g)h FA(of)h Fz(T)12 b(L)p FA(-p)r(erio)r(dic)515 2784 y(Sob)r(olev)27 b(v)n(ector-functions.)35 b(Here)28 b Fz(L)f FA(is)g(some)g(\014xed)h (in)n(teger.)639 2884 y(Let)g Fz(u)p FA(\()p Fz(t)p FA(\))g(b)r(e)g(a)f (smo)r(oth)g(solution)g(for)g(\(4.1\).)37 b(W)-7 b(e)28 b(set)g Fy(L)2435 2896 y Fw(t)2487 2884 y FA(=)23 b Fy(L)2632 2899 y Fw(u)p Fv(\()p Fw(t)p Fv(\))2780 2884 y FA(and)28 b Fy(A)3008 2896 y Fw(t)3060 2884 y FA(=)23 b Fy(A)3214 2899 y Fw(u)p Fv(\()p Fw(t)p Fv(\))3334 2884 y FA(.)515 3032 y FB(Lemma)35 b(4.1.)43 b Fp(L)l(et)33 b Fz(\037)1251 3044 y Fv(0)1320 3032 y Fy(2)e Fz(Z)1463 3044 y Fu(1)1567 3032 y Fp(b)l(e)k(a)f(smo)l(oth)h(eigenve)l(ctor)g(of)g Fy(L)2632 3044 y Fv(0)2669 3032 y Fp(,)h(i.e.,)h Fy(L)2962 3044 y Fv(0)3000 3032 y Fz(\037)3052 3044 y Fv(0)3120 3032 y FA(=)31 b Fz(\025\037)3316 3044 y Fv(0)3354 3032 y Fp(.)515 3132 y(L)l(et)e(us)g(assume)h(that)g(the)g(initial-value)h (pr)l(oblem)1517 3293 y FA(_)-42 b Fz(\037)23 b FA(=)g Fy(A)1727 3305 y Fw(t)1756 3293 y Fz(\037;)184 b(\037)p FA(\(0\))23 b(=)f Fz(\037)2335 3305 y Fv(0)2373 3293 y Fz(;)812 b FA(\(4.3\))515 3453 y Fp(has)30 b(a)g(unique)g(smo)l(oth)g (solution)g Fz(\037)p FA(\()p Fz(t)p FA(\))p Fp(.)39 b(Then)1586 3614 y Fy(L)1643 3626 y Fw(t)1672 3614 y Fz(\037)p FA(\()p Fz(t)p FA(\))24 b(=)e Fz(\025\037)p FA(\()p Fz(t)p FA(\))86 b Fy(8)p Fz(t:)899 b FA(\(4.4\))515 3775 y Fp(Pr)l(o)l(of.)43 b FA(Let)29 b(us)g(denote)g(the)g(left-hand)g (side)g(of)36 b(\(4.4\))28 b(b)n(y)h Fz(\030)t FA(\()p Fz(t)p FA(\),)h(the)f(righ)n(t-hand)f(side)h(|)515 3874 y(b)n(y)e Fz(\021)s FA(\()p Fz(t)p FA(\))h(and)g(calculate)f(their)g (deriv)-5 b(ativ)n(es.)36 b(W)-7 b(e)28 b(ha)n(v)n(e:)1160 4022 y Fz(d)p 1145 4059 V 1145 4135 a(dt)1242 4078 y(\030)f FA(=)1418 4022 y Fz(d)p 1403 4059 V 1403 4135 a(dt)1500 4078 y Fy(L)p Fz(\037)c FA(=)g([)p Fy(A)p Fz(;)14 b Fy(L)p FA(])p Fz(\037)19 b FA(+)f Fy(LA)p Fz(\037)23 b FA(=)g Fy(AL)p Fz(\037)h FA(=)e Fy(A)p Fz(\030)515 4262 y FA(and)1478 4328 y Fz(d)p 1463 4365 V 1463 4441 a(dt)1560 4384 y(\021)27 b FA(=)1740 4328 y Fz(d)p 1725 4365 V 1725 4441 a(dt)1822 4384 y(\025\037)d FA(=)e Fz(\025)p Fy(A)p Fz(\037)i FA(=)f Fz(A\021)s(:)515 4549 y FA(Th)n(us,)30 b(b)r(oth)g Fz(\030)t FA(\()p Fz(t)p FA(\))g(and)g Fz(\021)s FA(\()p Fz(t)p FA(\))g(solv)n(e)e(the)i(problem)g(\(4.3\))f(with)h Fz(\037)2564 4561 y Fv(0)2631 4549 y FA(replaced)f(b)n(y)g Fz(\025\037)3178 4561 y Fv(0)3245 4549 y FA(and)515 4648 y(coincide)e(b)n(y)g(the)h (uniqueness)g(assumption.)p 3318 4648 4 57 v 3322 4595 50 4 v 3322 4648 V 3372 4648 4 57 v 515 4692 1146 4 v 607 4746 a Fm(4)642 4770 y Fl(Due)h(to)g(a)f(deep)i(result)e(b)n(y)h (Kric)n(hev)n(er{Phong)g([KP98],)g(an)n(y)g(Lax-in)n(tegrable)h(PDE)e (is)g(a)g(Hamil-)515 4848 y(tonian)h(system.)44 b(The)29 b(corresp)r(onding)g(symplectic)f(structure)h(b)r(elongs)g(to)g(a)g (bigger)f(class)g(than)i(that)515 4927 y(describ)r(ed)j(in)g(Section)h (2.2,)h(so)e(to)g(apply)h(our)f(tec)n(hniques)h(w)n(e)g(need)f(to)h (assume)e(a)h(priori)f(that)i(the)515 5006 y(equation)25 b(has)f(the)g(form)e(\(4.1\))q(.)1905 5255 y FA(10)p eop PStoPSsaved restore %%Page: (10,11) 6 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 11 10 bop 639 523 a FA(Due)25 b(to)f(this)h(lemma)f(the)h(discrete)f (sp)r(ectrum)h(of)f(the)h(op)r(erator)d Fy(L)2752 535 y Fw(u)2821 523 y FA(is)i(an)g(in)n(tegral)f(of)515 623 y(motion)29 b(for)g(equation)f(\(4.1\).)42 b(In)29 b(particular,)g(a)g (set)g Fy(T)51 b FA(formed)29 b(b)n(y)g(all)g(smo)r(oth)g(v)n(ectors) 515 722 y Fz(u)22 b Fy(2)i Fz(X)733 734 y Fu(1)827 722 y FA(suc)n(h)g(that)g(the)g(eigen)n(v)-5 b(alues)23 b(of)h(the)h(op)r (erator)d Fy(L)2371 734 y Fw(u)2439 722 y FA(b)r(elong)h(to)h(a)g (\014xed)g(subset)g(of)515 822 y Fx(C)36 b Fy(\002)16 b Fx(C)36 b Fy(\002)16 b Fz(:)e(:)g(:)f FA(,)27 b(is)f(in)n(v)-5 b(arian)n(t)25 b(for)h(the)g(\015o)n(w)g(of)g(equation)g(\(4.1\))o(.)37 b(A)26 b(remark)-5 b(able)25 b(disco)n(v)n(ery)-7 b(,)515 922 y(made)30 b(b)n(y)h(No)n(vik)n(o)n(v)e([No)n(v74)n(])i(and)g(Lax)f ([Lax75)n(],)i(is)e(that)h(for)g(in)n(tegrable)e(Hamiltonian)515 1021 y(PDEs,)34 b(considered)f(on)h(\014nite)g(space-in)n(terv)-5 b(als)32 b(with)i(suitable)g(b)r(oundary)e(conditions,)515 1121 y(some)k(sets)h Fy(T)59 b FA(as)36 b(ab)r(o)n(v)n(e)g(are)h (\014nite)g(dimensional)g(symplectic)g(submanifolds)g Fy(T)3197 1091 y Fv(2)p Fw(n)3312 1121 y FA(of)515 1220 y(all)32 b(symplectic)h(spaces)e(\()p Fz(X)1409 1232 y Fw(s)1445 1220 y Fz(;)14 b(\013)1535 1232 y Fv(2)1572 1220 y FA(\),)34 b(and)f(restriction)f(of)g(equation)g(\(4.1\))h(to)f (an)n(y)g Fy(T)3212 1190 y Fv(2)p Fw(n)3323 1220 y FA(is)515 1320 y(an)f(in)n(tegrable)f(Hamiltonian)h(system.)48 b(Moreo)n(v)n(er,)30 b(for)h(some)g(in)n(tegrable)f(equations)h(it)515 1420 y(is)40 b(kno)n(wn)f(that)h(the)g(union)g(of)g(all)g(these)g (manifolds)g Fy(T)2391 1390 y Fv(2)p Fw(n)2509 1420 y FA(is)g(dense)g(in)g(ev)n(ery)f(space)515 1519 y Fz(X)584 1531 y Fw(s)619 1519 y FA(.)50 b(Solutions)32 b(that)g(\014ll)g(a)g (manifold)g Fy(T)1853 1489 y Fv(2)p Fw(n)1963 1519 y FA(are)f(called)h Fp(\014nite-gap)i(solutions)7 b FA(,)33 b(and)f(the)515 1619 y(manifold)g(itself)g({)f(a)g Fp(\014nite-gap)j (manifold)p FA(.)52 b(See)31 b(e.g.)49 b([DMN76,)32 b(ZMNP84)o(,)g(BBE) 3219 1589 y Fv(+)3273 1619 y FA(94)o(,)515 1719 y(Kuk00)n(].)515 1950 y Fs(4.2)112 b(Kortew)m(eg{de)37 b(V)-9 b(ries)36 b(equation)515 2103 y FA(The)27 b(KdV)h(equation)731 2333 y(_)-38 b Fz(u)23 b FA(=)884 2277 y(1)p 884 2314 42 4 v 884 2390 a(4)983 2277 y Fz(@)p 960 2314 97 4 v 960 2390 a(@)5 b(x)1066 2333 y FA(\()p Fz(u)1146 2345 y Fw(xx)1243 2333 y FA(+)18 b(3)p Fz(u)1416 2299 y Fv(2)1453 2333 y FA(\))p Fz(;)180 b(u)p FA(\()p Fz(t;)14 b(x)p FA(\))23 b Fy(\021)g Fz(u)p FA(\()p Fz(t;)14 b(x)19 b FA(+)f(2)p Fz(\031)s FA(\))p Fz(;)2505 2220 y Fq(Z)2588 2241 y Fv(2)p Fw(\031)2551 2409 y Fv(0)2680 2333 y Fz(u)c(dx)23 b Fy(\021)f FA(0)p Fz(;)201 b FA(\(4.5\))515 2567 y(tak)n(es)36 b(the)h(form)f(\(4.1\))h(in)g(the)g(symplectic)g(Hilb)r(ert)g(scale)f (\()p Fy(f)p Fz(X)2624 2579 y Fw(s)2660 2567 y Fy(g)p Fz(;)14 b(\013)2792 2579 y Fv(2)2867 2567 y FA(=)p 2970 2500 55 4 v 38 w Fz(J)22 b(du)i Fy(^)h Fz(du)p FA(\),)515 2666 y(where)32 b Fz(X)829 2678 y Fw(s)896 2666 y FA(is)h(the)f(Sob)r (olev)g(space)g Fz(H)1749 2636 y Fw(s)1784 2666 y FA(\()p Fz(S)1872 2636 y Fv(1)1909 2666 y FA(\))1941 2678 y Fv(0)2011 2666 y FA(and)h Fz(J)8 b(u)30 b FA(=)h(\()p Fz(@)5 b(=@)g(x)p FA(\))p Fz(u)p FA(,)33 b(see)f(Example)g(2.7.)515 2766 y(Due)c(to)h(Lax)e(himself,)i(this)f(equation)g(is)g(Lax-in)n(tegrable) e(and)i(the)h(corresp)r(onding)d(Lax)515 2866 y(pair)h(is)1048 3014 y Fy(L)1105 3026 y Fw(u)1172 3014 y FA(=)22 b Fy(\000)1357 2958 y Fz(@)1406 2928 y Fv(2)p 1334 2995 134 4 v 1334 3071 a Fz(@)5 b(x)1430 3047 y Fv(2)1495 3014 y Fy(\000)18 b Fz(u;)180 b Fy(A)1895 3026 y Fw(u)1962 3014 y FA(=)2083 2958 y Fz(@)2132 2928 y Fv(3)p 2059 2995 V 2059 3071 a Fz(@)5 b(x)2155 3047 y Fv(3)2221 3014 y FA(+)2314 2958 y(3)p 2314 2995 42 4 v 2314 3071 a(2)2379 3014 y Fz(u)2474 2958 y(@)p 2451 2995 97 4 v 2451 3071 a(@)g(x)2575 3014 y FA(+)2668 2958 y(3)p 2668 2995 42 4 v 2668 3071 a(4)2734 3014 y Fz(u)2782 3026 y Fw(x)2823 3014 y Fz(:)515 3187 y FA(T)-7 b(aking)22 b(for)g Fy(f)p Fz(Z)1008 3199 y Fw(s)1043 3187 y Fy(g)g FA(the)h(Sob)r(olev)f(scale)g(of)h(4)p Fz(\031)s FA(-p)r(erio)r(dic)f(functions)h(and)g(applying)f(Lemma)515 3287 y(4.1)d(w)n(e)h(obtain)f(that)i(smo)r(oth)e(4)p Fz(\031)s FA(-p)r(erio)r(dic)h(sp)r(ectrum)g(of)g(the)h(op)r(erator)d Fy(L)2857 3299 y Fw(u)2921 3287 y FA(is)i(an)f(in)n(tegral)515 3386 y(of)27 b(motion.)37 b(It)28 b(is)f(w)n(ell)h(kno)n(wn)f(that)g (the)h(sp)r(ectrum)g(of)g Fy(L)2365 3398 y Fw(u)2436 3386 y FA(is)g(formed)f(b)n(y)g(eigen)n(v)-5 b(alues)1290 3562 y Fz(\025)1338 3574 y Fv(0)1399 3562 y Fz(<)22 b(\025)1534 3574 y Fv(1)1595 3562 y Fy(\024)h Fz(\025)1731 3574 y Fv(2)1791 3562 y Fz(<)g(\025)1927 3574 y Fv(3)1988 3562 y Fy(\024)f Fz(\025)2123 3574 y Fv(4)2184 3562 y Fz(<)h Fy(\001)14 b(\001)g(\001)23 b(\045)g(1)p Fz(;)515 3738 y FA(and)34 b(that)g(the)h(corresp)r(onding)d(eigenfunctions)i(are)f (smo)r(oth,)i(pro)n(vided)e(that)i(the)f(p)r(o-)515 3837 y(ten)n(tial)27 b Fz(u)h FA(is.)36 b(Let)28 b(us)g(tak)n(e)e(an)n(y)h (in)n(teger)g Fz(n)p FA(-v)n(ector)f FB(V)q FA(,)1175 4013 y FB(V)f FA(=)e(\()p Fz(V)1440 4025 y Fv(1)1478 4013 y Fz(;)14 b(:)g(:)g(:)f(;)h(V)1710 4025 y Fw(n)1756 4013 y FA(\))23 b Fy(2)h Fx(N)1943 3979 y Fw(n)1995 4013 y Fz(;)180 b(V)2246 4025 y Fv(1)2306 4013 y Fz(<)23 b Fy(\001)14 b(\001)g(\001)23 b Fz(<)g(V)2650 4025 y Fw(n)2695 4013 y Fz(:)515 4189 y FA(Denoting)k(\001)942 4201 y Fw(j)1001 4189 y FA(=)22 b Fz(\025)1136 4201 y Fv(2)p Fw(j)1223 4189 y Fy(\000)c Fz(\025)1354 4201 y Fv(2)p Fw(j)s Fu(\000)p Fv(1)1531 4189 y Fy(\025)k FA(0,)28 b Fz(j)g FA(=)22 b(1)p Fz(;)14 b FA(2)p Fz(;)g(:)g(:)g(:)f FA(,)28 b(w)n(e)f(de\014ne)h(the)g(set)f Fy(T)2879 4159 y Fv(2)p Fw(n)2858 4212 y Fi(V)2985 4189 y FA(as)1160 4365 y Fy(T)1226 4330 y Fv(2)p Fw(n)1205 4385 y Fi(V)1327 4365 y FA(=)c Fy(f)p Fz(u)p FA(\()p Fz(x)p FA(\))g Fy(j)g FA(\001)1754 4377 y Fw(j)1812 4365 y Fy(6)p FA(=)g(0)k(i\013)42 b Fz(j)28 b Fy(2)c(f)p Fz(V)2313 4377 y Fv(1)2350 4365 y Fz(;)14 b(:)g(:)g(:)f(;)h(V)2582 4377 y Fw(n)2628 4365 y Fy(gg)p Fz(:)515 4540 y FA(Clearly)33 b Fy(T)876 4510 y Fv(2)p Fw(n)855 4563 y Fi(V)988 4540 y FA(equals)h(to)g(the)g(union)g Fy(T)1807 4510 y Fv(2)p Fw(n)1786 4563 y Fi(V)1919 4540 y FA(=)2018 4478 y Fq(S)2087 4565 y Fw(r)r Fu(2)p Fo(R)2212 4545 y Fn(n)2212 4584 y Ft(+)2271 4540 y Fz(T)2332 4510 y Fw(n)2320 4563 y Fi(V)2381 4540 y FA(\()p Fz(r)r FA(\))p Fz(;)i FA(where)d Fx(R)2843 4510 y Fw(n)2843 4561 y Fv(+)2938 4540 y FA(=)g Fy(f)p Fz(r)j Fy(j)e Fz(r)3245 4552 y Fw(j)3314 4540 y Fz(>)515 4662 y FA(0)22 b Fy(8)p Fz(j)5 b Fy(g)27 b FA(and)1296 4761 y Fz(T)1357 4727 y Fw(n)1345 4782 y Fi(V)1406 4761 y FA(\()p Fz(r)r FA(\))e(=)d Fy(f)p Fz(u)p FA(\()p Fz(x)p FA(\))h Fy(2)h(T)1990 4727 y Fv(2)p Fw(n)1969 4782 y Fi(V)2091 4761 y Fy(j)g FA(\001)2207 4773 y Fw(j)2265 4761 y FA(=)e Fz(r)2389 4773 y Fw(j)2448 4761 y Fy(8)p Fz(j)5 b Fy(g)p Fz(:)515 4907 y FA(Since)32 b(the)h(4)p Fz(\031)s FA(-p)r(erio)r(dic)e(sp)r(ectrum)i Fy(f)p Fz(\025)1784 4919 y Fw(j)1819 4907 y Fy(g)f FA(is)g(an)g(in)n (tegral)f(of)h(motion)g(for)g(\(KdV\),)h(then)515 5006 y(the)24 b(sets)f Fz(T)873 4976 y Fw(n)861 5029 y Fi(V)922 5006 y FA(\()p Fz(r)r FA(\))i(are)d(in)n(v)-5 b(arian)n(t)23 b(for)g(the)h(KdV-\015o)n(w.)34 b(Due)24 b(to)f(the)h(classical)e (theory)h(of)h(the)1905 5255 y(11)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 12 11 bop 515 523 a FA(Sturm{Liouville)24 b(op)r(erator)f Fy(L)1509 535 y Fw(u)1552 523 y FA(,)j(the)f(set)f Fy(T)1934 493 y Fv(2)p Fw(n)1912 546 y Fi(V)2037 523 y FA(is)g(a)g(smo)r(oth)h (submanifold)f(of)h(an)n(y)f(space)515 623 y Fz(X)584 635 y Fw(s)619 623 y FA(,)k(foliated)f(to)h(the)g(smo)r(oth)f Fz(n)p FA(-tori)g Fz(T)1801 593 y Fw(n)1789 646 y Fi(V)1850 623 y FA(\()p Fz(r)r FA(\).)38 b(F)-7 b(or)27 b(all)g(these)h(results)f (see)g(e.g.)36 b([KP03)o(].)639 722 y(Due)d(to)f(No)n(vik)n(o)n(v)e (and)i(Lax,)g(there)g(exist)g(an)g(analytic)g(map)g(\010)e(=)g(\010) 2901 734 y Fi(V)2972 722 y FA(:)f Fy(f)p FA(\()p Fz(r)n(;)14 b(\030)t FA(\))p Fy(g)30 b FA(=)515 822 y Fx(R)569 792 y Fw(n)569 842 y Fv(+)645 822 y Fy(\002)15 b Fx(T)781 792 y Fw(n)848 822 y Fy(!)23 b Fz(X)1023 834 y Fw(s)1085 822 y FA(\()p Fz(s)j FA(is)g(an)n(y)f(in)n(teger\),)h(and)g(an)f (analytic)h(function)g Fz(h)d FA(=)g Fz(h)2866 792 y Fw(n)2911 822 y FA(\()p Fz(r)r FA(\))k(suc)n(h)f(that)515 922 y Fz(T)576 891 y Fw(n)564 945 y Fi(V)625 922 y FA(\()p Fz(r)r FA(\))e(=)f(\010\()p Fy(f)p Fz(r)r Fy(g)14 b(\002)g Fx(T)1204 891 y Fw(n)1248 922 y FA(\),)27 b(and)e(for)g(an)n(y)g Fz(\030)1805 934 y Fv(0)1865 922 y Fy(2)f Fx(T)2000 891 y Fw(n)2070 922 y FA(the)i(curv)n(e)e Fz(u)p FA(\()p Fz(t)p FA(\))f(=)g(\010\()p Fz(r)n(;)14 b(\030)2883 934 y Fv(0)2935 922 y FA(+)g Fz(t)p Fy(r)p Fz(h)p FA(\()p Fz(r)r FA(\)\))27 b(is)515 1021 y(a)g(smo)r(oth)h(solution)f(for)g (\(4.5\).)37 b(As)28 b(a)f(function)i(of)e Fz(t)p FA(,)h(this)g (solution)g(is)f(quasip)r(erio)r(dic.)3342 991 y Fv(5)515 1121 y FA(The)k(celebrated)g(Its{Matv)n(eev)g(form)n(ula)f(explicitly)i (represen)n(ts)f(\010)g(in)h(terms)g(of)f(theta-)515 1220 y(functions,)d(see)f(in)h([DMN76,)f(Dub81,)h(BBE)1950 1190 y Fv(+)2004 1220 y FA(94)o(,)g(Kuk00)n(].)515 1453 y Fs(4.3)112 b(Other)37 b(examples)515 1606 y Fp(Sine-Gor)l(don)p FA(.)g(The)28 b(Sine-Gordon)f(\(SG\))h(equation)f(on)g(the)h(circle) 1085 1789 y(\177)-48 b Fz(u)23 b FA(=)g Fz(u)1286 1801 y Fw(xx)1365 1789 y FA(\()p Fz(t;)14 b(x)p FA(\))19 b Fy(\000)f FA(sin)c Fz(u)p FA(\()p Fz(t;)g(x)p FA(\))p Fz(;)180 b(x)24 b Fy(2)f Fz(S)2395 1754 y Fv(1)2455 1789 y FA(=)g Fx(R)p Fz(=)p FA(2)p Fz(\031)s Fx(Z)p Fz(;)515 1971 y FA(is)k(another)g(example)g(of)h(a)f(Lax-in)n(tegrable)e(HPDE.) 639 2071 y(First)41 b(the)g(equation)f(has)g(to)h(b)r(e)g(written)g(in) g(a)f(Hamiltonian)h(form.)75 b(The)41 b(most)515 2171 y(straigh)n(tforw)n(ard)24 b(w)n(as)j(to)g(do)h(this)g(is)f(to)h(write) f(\(SG\))h(as)f(the)h(system)1321 2353 y(_)-38 b Fz(u)23 b FA(=)g Fy(\000)p Fz(v)s(;)192 b FA(_)-36 b Fz(v)27 b FA(=)22 b Fy(\000)p Fz(u)2042 2365 y Fw(xx)2139 2353 y FA(+)c(sin)c Fz(u)p FA(\()p Fz(t;)g(x)p FA(\))p Fz(:)515 2536 y FA(One)27 b(immediately)h(sees)f(that)h(this)g(system)f(is)h(a)f (semilinear)g(Hamiltonian)h(equation)f(in)515 2636 y(the)f(symplectic)g (scale)f(\()p Fy(f)p Fz(X)1403 2648 y Fw(s)1461 2636 y FA(=)d Fz(H)1624 2605 y Fw(s)1660 2636 y FA(\()p Fz(S)5 b FA(\))15 b Fy(\002)f Fz(H)1950 2605 y Fw(s)1985 2636 y FA(\()p Fz(S)5 b FA(\))p Fy(g)p Fz(;)14 b(\013)2237 2648 y Fv(2)2297 2636 y FA(=)p 2385 2569 55 4 v 23 w Fz(J)22 b(d\021)c Fy(^)d Fz(d\021)s FA(\),)27 b(where)e Fz(\021)h FA(=)d(\()p Fz(u;)14 b(v)s FA(\))515 2735 y(and)27 b Fz(J)8 b FA(\()p Fz(u;)14 b(v)s FA(\))23 b(=)g(\()p Fy(\000)p Fz(v)s(;)14 b(u)p FA(\).)639 2835 y(No)n(w)32 b(w)n(e)g(deriv)n(e)f(another)g(Hamiltonian)h(form)g(of)g(\(SG\),)i (more)d(con)n(v)n(enien)n(t)g(for)h(its)515 2934 y(analysis.)47 b(T)-7 b(o)31 b(do)g(this)h(w)n(e)f(consider)f(the)i(shifted)g(Sob)r (olev)f(scale)f Fy(f)p Fz(X)2782 2946 y Fw(s)2847 2934 y FA(=)f Fz(H)3017 2904 y Fw(s)p Fv(+1)3136 2934 y FA(\()p Fz(S)3224 2904 y Fv(1)3261 2934 y FA(\))21 b Fy(\002)515 3034 y Fz(H)591 3004 y Fw(s)p Fv(+1)710 3034 y FA(\()p Fz(S)798 3004 y Fv(1)835 3034 y FA(\))p Fy(g)p FA(,)28 b(where)f(the)h(space)f Fz(X)1634 3046 y Fv(0)1698 3034 y FA(is)h(giv)n(en)f(the)h(scalar)e(pro)r(duct)1312 3259 y Fy(h)p Fz(\030)1380 3271 y Fv(1)1418 3259 y Fz(;)14 b(\030)1491 3271 y Fv(2)1529 3259 y Fy(i)23 b FA(=)1672 3146 y Fq(Z)1718 3335 y Fw(S)1762 3318 y Ft(1)1798 3259 y FA(\()p Fz(\030)1870 3225 y Fu(0)1866 3280 y Fv(1)p Fw(x)1960 3259 y Fy(\001)c Fz(\030)2042 3225 y Fu(0)2038 3280 y Fv(2)p Fw(x)2131 3259 y FA(+)f Fz(\030)2250 3271 y Fv(1)2306 3259 y Fy(\001)h Fz(\030)2384 3271 y Fv(2)2422 3259 y FA(\))14 b Fz(dx;)515 3488 y FA(and)30 b(an)n(y)f(space)h Fz(X)1132 3500 y Fw(s)1198 3488 y FA({)g(the)g(pro)r(duct)h Fy(h)p Fz(\030)1798 3500 y Fv(1)1836 3488 y Fz(;)14 b(\030)1909 3500 y Fv(2)1946 3488 y Fy(i)1978 3500 y Fw(s)2041 3488 y FA(=)28 b Fy(h)p Fz(A)2228 3458 y Fw(s)2264 3488 y Fz(\030)2300 3500 y Fv(1)2337 3488 y Fz(;)14 b(\030)2410 3500 y Fv(2)2448 3488 y Fy(i)p FA(.)45 b(Here)30 b Fz(A)h FA(is)f(the)h(op)r(erator)515 3588 y Fz(A)23 b FA(=)g Fy(\000)p Fz(@)802 3558 y Fv(2)838 3588 y Fz(=@)5 b(x)976 3558 y Fv(2)1026 3588 y FA(+)13 b(1.)35 b(Ob)n(viously)-7 b(,)25 b Fz(A)g FA(de\014nes)g(a)f(selfadjoin)n(t)h(automorphism)f(of)h (the)g(scale)515 3688 y(of)e(order)e(one.)35 b(The)23 b(op)r(erator)e Fz(J)8 b FA(\()p Fz(u;)14 b(w)r FA(\))24 b(=)f(\()p Fy(\000)1969 3617 y(p)p 2038 3617 63 4 v 71 x Fz(A)14 b(w)r(;)2212 3617 y Fy(p)p 2281 3617 V 71 x Fz(A)h(u)p FA(\))22 b(de\014nes)h(an)g(an)n(ti-selfadjoin)n(t)515 3787 y(automorphism)d(of)i(the)f(same)g(order.)34 b(W)-7 b(e)21 b(pro)n(vide)g(the)h(scale)e(with)i(the)g(symplectic)f(form)515 3887 y Fz(\014)562 3899 y Fv(2)622 3887 y FA(=)p 710 3820 55 4 v 23 w Fz(J)g(d\030)i Fy(^)c Fz(d\030)t FA(.)37 b(W)-7 b(e)28 b(note)g(that)g(\(SG\))g(can)f(b)r(e)h(written)g(as)f (the)h(system)1144 4080 y(_)-38 b Fz(u)23 b FA(=)f Fy(\000)1352 4006 y(p)p 1421 4006 63 4 v 74 x Fz(A)14 b(w)r(;)207 b FA(_)-49 b Fz(w)26 b FA(=)1934 4006 y Fy(p)p 2003 4006 V 74 x Fz(A)15 b FA(\()p Fz(u)j FA(+)g Fz(A)2323 4046 y Fu(\000)p Fv(1)2412 4080 y Fz(f)2462 4046 y Fu(0)2485 4080 y FA(\()p Fz(u)p FA(\()p Fz(x)p FA(\)\)\))p Fz(;)445 b FA(\(4.6\))515 4263 y(where)31 b Fz(f)9 b FA(\()p Fz(u)p FA(\))29 b(=)g Fy(\000)14 b FA(cos)e Fz(u)20 b Fy(\000)1411 4230 y Fv(1)p 1411 4244 34 4 v 1411 4291 a(2)1454 4263 y Fz(u)1502 4232 y Fv(2)1539 4263 y FA(,)32 b(and)f(that)h(\(4.6\))f (is)g(a)g(semilinear)g(Hamiltonian)g(equa-)515 4362 y(tion)j(in)g(the)h (symplectic)f(scale)f(as)g(ab)r(o)n(v)n(e)g(with)i(the)f(Hamiltonian)g Fz(H)7 b FA(\()p Fz(\030)t FA(\))34 b(=)3067 4330 y Fv(1)p 3067 4344 V 3067 4391 a(2)3110 4362 y Fy(h)p Fz(\030)t(;)14 b(\030)t Fy(i)23 b FA(+)515 4395 y Fq(R)584 4462 y Fz(f)9 b FA(\()p Fz(u)p FA(\()p Fz(x)p FA(\)\))14 b Fz(dx)p FA(,)29 b Fz(\030)e FA(=)c(\()p Fz(u;)14 b(w)r FA(\).)639 4561 y(Let)37 b(us)g(denote)f(b)n(y)h Fz(X)1390 4531 y Fw(o)1383 4582 y(s)1463 4561 y FA(\()p Fz(X)1571 4531 y Fw(e)1564 4582 y(s)1606 4561 y FA(\))g(subspaces)f(of)h Fz(X)2237 4573 y Fw(s)2309 4561 y FA(formed)f(b)n(y)h(o)r(dd)g(\(ev)n (en\))f(v)n(ector)515 4661 y(functions)23 b Fz(\030)t FA(\()p Fz(x)p FA(\).)37 b(Then)23 b(\()p Fy(f)p Fz(X)1441 4631 y Fw(o)1434 4682 y(s)1477 4661 y Fy(g)p Fz(;)14 b(\014)1603 4673 y Fv(2)1640 4661 y FA(\))23 b(and)g(\()p Fy(f)p Fz(X)2002 4631 y Fw(e)1995 4682 y(s)2037 4661 y Fy(g)p Fz(;)14 b(\014)2163 4673 y Fv(2)2200 4661 y FA(\))23 b(are)g(symplectic)g(sub-scales)f(of)h(the)515 4761 y(scale)31 b(ab)r(o)n(v)n(e.)51 b(The)33 b(space)e Fz(X)1479 4731 y Fw(o)1472 4781 y(s)1548 4761 y FA(and)i Fz(X)1791 4731 y Fw(e)1784 4781 y(s)1858 4761 y FA(\(with)h Fz(s)d Fy(\025)g FA(0\))h(are)g(in)n(v)-5 b(arian)n(t)31 b(for)h(the)h(\015o)n(w)f(of)p 515 4814 1146 4 v 607 4868 a Fm(5)642 4891 y Fl(A)24 b(con)n(tin)n(uous)i(curv)n(e)f Fk(u)c Fl(:)f Fh(R)h Fj(!)g Fk(X)30 b Fl(is)23 b(quasip)r(erio)r(dic)i (if)e(there)j(exist)e Fk(n)d Fj(2)g Fh(N)p Fl(,)28 b Fk(\036)21 b Fj(2)g Fh(T)2934 4868 y Fg(n)2971 4891 y Fl(,)j Fk(!)f Fj(2)e Fh(R)3198 4868 y Fg(n)3265 4891 y Fl(and)515 4970 y(a)j(con)n(tin)n(uous)h(map)e Fk(U)j Fl(:)19 b Fh(T)1247 4946 y Fg(n)1304 4970 y Fj(!)g Fk(X)29 b Fl(suc)n(h)24 b(that)h Fk(u)p Fl(\()p Fk(t)p Fl(\))20 b(=)g Fk(U)7 b Fl(\()p Fk(\036)16 b Fl(+)f Fk(t!)r Fl(\).)1905 5255 y FA(12)p eop PStoPSsaved restore %%Page: (12,13) 7 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 13 12 bop 515 523 a FA(equation)31 b(\(4.6\))o(.)49 b(The)32 b(restricted)f(\015o)n(ws)g(corresp)r(ond)f(to)h(the)h(SG)g(equation)f (under)h(the)515 623 y(o)r(dd)27 b(p)r(erio)r(dic)h(and)f(ev)n(en)g(p)r (erio)r(dic)h(b)r(oundary)f(conditions,)g(resp)r(ectiv)n(ely)-7 b(.)639 771 y(The)35 b(SG)h(equation)e(is)h(Lax-in)n(tegrable)d(under)j (p)r(erio)r(dic,)i(o)r(dd)e(p)r(erio)r(dic)g(and)g(ev)n(en)515 871 y(p)r(erio)r(dic)c(b)r(oundary)f(conditions.)48 b(That)31 b(is,)h(equation)f(\(4.6\))g(is)g(Lax-in)n(tegrable)e(in)j(the)515 970 y(all)27 b(three)g(symplectic)h(scales)f(de\014ned)h(ab)r(o)n(v)n (e.)35 b(See)28 b([BBE)2382 940 y Fv(+)2436 970 y FA(94)o(,)g(Kuk00)n (].)639 1119 y Fp(Zakhar)l(ov{Shab)l(at)36 b(e)l(quation.)48 b FA(Let)32 b(us)f(tak)n(e)f(the)i(symplectic)f(Hilb)r(ert)h(scale)e (\()p Fz(X)3250 1131 y Fw(s)3314 1119 y FA(=)515 1218 y Fz(H)591 1188 y Fw(s)626 1218 y FA(\()p Fz(S)714 1188 y Fv(1)751 1218 y Fz(;)14 b Fx(C)h FA(\),)p 946 1152 55 4 v 49 w Fz(J)22 b(du)k Fy(^)g Fz(du)p FA(\))40 b(as)e(in)i(the)g (Example)e(2.8.)72 b(The)39 b(defo)r(cusing)g(and)g(fo)r(cusing)515 1318 y(Zakharo)n(v{Shabat)24 b(equations)1294 1498 y(_)-38 b Fz(u)23 b FA(=)g Fz(i)p FA(\()p Fy(\000)p Fz(u)1612 1510 y Fw(xx)1708 1498 y FA(+)18 b Fz(mu)g Fy(\006)g Fz(\015)5 b Fy(j)p Fz(u)p Fy(j)2155 1464 y Fv(2)2192 1498 y Fz(u)p FA(\))p Fz(;)97 b(\015)27 b(>)c FA(0)p Fz(;)593 b FA(\(4.7\))515 1678 y(b)r(oth)28 b(are)e(Lax-in)n(tegrable,) g(see)h([ZMNP84)o(,)g(BBE)2134 1648 y Fv(+)2189 1678 y FA(94)o(].)515 1952 y FC(5)134 b(KAM)45 b(for)g(PDEs)515 2134 y FA(In)37 b(this)g(section)f(w)n(e)h(discuss)f(the)h(KAM)g(for)f (PDEs)g(theory)-7 b(.)64 b(W)-7 b(e)37 b(co)n(v)n(er)e(all)i(relev)-5 b(an)n(t)515 2234 y(topics,)23 b(except)f(the)h(theory)e(of)h(time-p)r (erio)r(dic)h(solutions)e(of)h(Hamiltonian)g(PDEs)g(\(since,)515 2333 y(\014rstly)-7 b(,)41 b(this)e(theory)f(is)h(an)f(extensiv)n(e)g (sub)5 b(ject)39 b(and,)j(secondly)-7 b(,)41 b(man)n(y)d(results)g (there)515 2433 y(ha)n(v)n(e)26 b(b)r(een)h(recen)n(tly)g(pro)n(v)n(ed) e(and)i(re-pro)n(v)n(ed)e(without)j(the)f(KAM-mac)n(hinery)-7 b(,)26 b(e.g.)36 b(see)515 2533 y([Bam00)n(]\).)71 b(W)-7 b(e)38 b(a)n(v)n(oid)g(completely)g(the)h(classical)e (\014nite-dimensional)h(KAM-theory)515 2632 y(whic)n(h)24 b(deals)f(with)h(time-quasip)r(erio)r(dic)g(solutions)f(of)h (\014nite-dimensional)f(Hamiltonian)515 2732 y(systems)k(and)g(instead) h(refer)f(the)h(reader)e(to)h(the)h(recen)n(t)f(surv)n(ey)f([Sev03)o (].)515 2964 y Fs(5.1)112 b(An)37 b(abstract)h(KAM-theorem)515 3117 y FA(Let)30 b(\()p Fy(f)p Fz(X)809 3129 y Fw(s)844 3117 y Fy(g)p Fz(;)14 b(\013)976 3129 y Fv(2)1040 3117 y FA(=)1152 3096 y(\026)1132 3117 y Fz(J)8 b(du)20 b Fy(^)g Fz(du)p FA(\))30 b(b)r(e)h(a)f(symplectic)g(Hilb)r(ert)h(scale,) f Fy(\000)p Fz(d)2746 3129 y Fw(J)2819 3117 y FA(=)g(ord)3068 3096 y(\026)3049 3117 y Fz(J)k Fy(\024)27 b FA(0;)k Fz(A)515 3217 y FA(b)r(e)h(an)g(op)r(erator)e(whic)n(h)i(de\014nes)g(a)f (selfadjoin)n(t)h(automorphism)f(of)h(the)g(scale)g(of)f(order)515 3316 y Fz(d)558 3328 y Fw(A)643 3316 y Fy(\025)f(\000)p Fz(d)846 3328 y Fw(J)925 3316 y FA(and)i Fz(H)39 b FA(b)r(e)33 b(a)e(F)-7 b(r)n(\023)-39 b(ec)n(het{analytic)30 b(functional)j(on)f Fz(X)2603 3328 y Fw(d)2638 3336 y Ft(0)2673 3316 y FA(,)i Fz(d)2773 3328 y Fv(0)2841 3316 y Fy(\025)d FA(0,)i(suc)n(h)f(that)515 3416 y(ord)14 b Fy(r)p Fz(H)29 b FA(=)22 b Fz(d)947 3428 y Fw(H)1034 3416 y Fz(<)g(d)1164 3428 y Fw(A)1219 3416 y FA(:)1554 3516 y Fy(r)p Fz(H)30 b FA(:)23 b Fz(X)1837 3528 y Fw(d)1872 3536 y Ft(0)1931 3516 y Fy(!)g Fz(X)2106 3528 y Fw(d)2141 3536 y Ft(0)2173 3528 y Fu(\000)p Fw(d)2260 3536 y Fn(H)2317 3516 y Fz(:)515 3664 y FA(W)-7 b(e)24 b(assume)f(that)i Fz(d)1157 3676 y Fw(A)1234 3664 y Fy(\024)e FA(2)p Fz(d)1407 3676 y Fv(0)1443 3664 y FA(,)i(so)f(the)g(quadratic)f (form)2299 3631 y Fv(1)p 2299 3645 34 4 v 2299 3692 a(2)2342 3664 y Fy(h)p Fz(Au;)14 b(u)p Fy(i)24 b FA(is)g(w)n(ell)g(de\014ned)g (on)g(the)515 3763 y(space)j Fz(X)806 3775 y Fw(d)841 3783 y Ft(0)876 3763 y FA(.)639 3863 y(In)35 b(this)g(section)f(w)n(e)g (consider)g(the)g(quasilinear)f(Hamiltonian)i(equation)f(with)h(the)515 3962 y(Hamiltonian)27 b Fz(H)1061 3974 y Fw(")1097 3962 y FA(\()p Fz(u)p FA(\))c(=)1329 3930 y Fv(1)p 1329 3944 V 1329 3991 a(2)1372 3962 y Fy(h)p Fz(Au;)14 b(u)p Fy(i)19 b FA(+)f Fz("H)7 b FA(\()p Fz(u)p FA(\):)1432 4143 y(_)-37 b Fz(u)o FA(\()p Fz(t)p FA(\))24 b(=)f Fz(J)8 b FA(\()p Fz(Au)p FA(\()p Fz(t)p FA(\))19 b(+)f Fz(")p Fy(r)p Fz(H)7 b FA(\()p Fz(u)p FA(\()p Fz(t)p FA(\)\))p Fz(:)732 b FA(\(5.1\))515 4323 y(W)-7 b(e)28 b(assume)e(that)i(the)g(scale)e Fy(f)p Fz(X)1576 4335 y Fw(s)1611 4323 y Fy(g)h FA(admits)g(a)g(basis)g Fy(f)p Fz(')2321 4335 y Fw(k)2362 4323 y Fz(;)14 b(k)26 b Fy(2)d Fx(Z)2607 4335 y Fv(0)2662 4323 y FA(=)f Fx(Z)p Fy(n)o(f)p FA(0)o Fy(gg)f FA(suc)n(h)27 b(that)1225 4503 y Fz(A')1341 4467 y Fu(\006)1341 4526 y Fw(j)1420 4503 y FA(=)c Fz(\025)1556 4469 y Fw(A)1556 4523 y(j)1611 4503 y Fz(')1665 4467 y Fu(\006)1665 4526 y Fw(j)1721 4503 y Fz(;)60 b(J)8 b(')1912 4467 y Fu(\006)1912 4526 y Fw(j)1991 4503 y FA(=)23 b Fy(\007)p Fz(\025)2192 4469 y Fw(J)2192 4523 y(j)2238 4503 y Fz(')2292 4467 y Fu(\006)2292 4526 y Fw(j)2409 4503 y Fy(8)p Fz(j)k Fy(\025)22 b FA(1)p Fz(;)539 b FA(\(5.2\))515 4696 y(with)24 b(some)g(real)f(n)n(um)n(b)r (ers)g Fz(\025)1442 4666 y Fw(J)1442 4718 y(j)1489 4696 y FA(,)i Fz(\025)1585 4666 y Fw(A)1585 4718 y(j)1640 4696 y FA(.)35 b(In)25 b(particular,)e(the)i(sp)r(ectrum)f(of)g(the)g (op)r(erator)f Fz(J)8 b(A)515 4807 y FA(is)27 b Fy(f\006)p Fz(i\025)782 4819 y Fw(j)839 4807 y Fy(j)c Fz(\025)933 4819 y Fw(j)992 4807 y FA(=)f Fz(\025)1127 4777 y Fw(J)1127 4829 y(j)1174 4807 y Fz(\025)1222 4777 y Fw(A)1222 4829 y(j)1277 4807 y Fy(g)p FA(.)36 b(The)28 b(n)n(um)n(b)r(ers)f Fz(\025)1932 4819 y Fw(j)1994 4807 y FA(are)g(called)g(the)h Fp(fr)l(e)l(quencies)35 b FA(of)27 b(the)h(linear)515 4907 y(system)1789 5006 y(_)-38 b Fz(u)23 b FA(=)g Fz(J)8 b(Au:)1088 b FA(\(5.3\))1905 5255 y(13)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 14 13 bop 639 523 a FA(Let)28 b(us)g(\014x)f(an)n(y)g Fz(n)c Fy(\025)g FA(1.)36 b(Then)28 b(the)g(2)p Fz(n)p FA(-dimensional)e(linear)h(space)1556 682 y(span)p Fy(f)p Fz(')1819 647 y Fu(\006)1819 706 y Fw(j)1898 682 y Fy(j)c FA(1)g Fy(\024)f Fz(j)28 b Fy(\024)23 b Fz(n)p Fy(g)870 b FA(\(5.4\))515 842 y(is)27 b(in)n(v)-5 b(arian)n(t)27 b(for)g(the)h(equation)f(\(5.3\))g(and)h(is)f(foliated)g(to)h(the)g(in) n(v)-5 b(arian)n(t)26 b(tori)984 1101 y Fz(T)1045 1067 y Fw(n)1113 1101 y FA(=)c Fz(T)1261 1067 y Fw(n)1306 1101 y FA(\()p Fz(I)7 b FA(\))23 b(=)1524 931 y Fq(8)1524 1005 y(<)1524 1155 y(:)1637 997 y Fw(n)1598 1022 y Fq(X)1600 1199 y Fw(j)s Fv(=1)1731 1101 y Fz(u)1779 1065 y Fu(\006)1779 1124 y Fw(j)1835 1101 y Fz(')1889 1065 y Fu(\006)1889 1124 y Fw(j)1968 1101 y Fy(j)h Fz(u)2063 1113 y Fw(j)2097 1067 y Fv(+)2152 1051 y(2)2208 1101 y FA(+)18 b Fz(u)2339 1113 y Fw(j)2373 1067 y Fu(\000)2429 1051 y Fv(2)2490 1101 y FA(=)k(2)p Fz(I)2655 1113 y Fw(j)2713 1101 y Fy(8)p Fz(j)2799 931 y Fq(9)2799 1005 y(=)2799 1155 y(;)2886 1101 y Fz(:)299 b FA(\(5.5\))515 1360 y(If)36 b Fz(I)43 b Fy(2)37 b Fx(R)831 1330 y Fw(n)831 1381 y Fv(+)893 1360 y FA(,)g(then)g Fz(T)1212 1330 y Fw(n)1256 1360 y FA(\()p Fz(I)7 b FA(\))36 b(is)g(an)f Fz(n)p FA(-torus.)60 b(Pro)n(viding)34 b(it)i(with)g(the)g(co)r(ordinates)f Fz(q)k FA(=)515 1460 y(\()p Fz(q)584 1472 y Fv(1)621 1460 y Fz(;)14 b(:)g(:)g(:)g(;)g(q)843 1472 y Fw(n)888 1460 y FA(\),)34 b(where)d Fz(q)1258 1472 y Fw(j)1323 1460 y FA(=)f(Arg)q(\()p Fz(u)1636 1424 y Fv(+)1636 1483 y Fw(j)1712 1460 y FA(+)21 b Fz(iu)1875 1424 y Fu(\000)1875 1483 y Fw(j)1930 1460 y FA(\),)34 b(w)n(e)d(see)h(that)g(equation)g (\(5.3\))f(de\014nes)h(on)515 1559 y Fz(T)576 1529 y Fw(n)620 1559 y FA(\()p Fz(I)7 b FA(\))28 b(the)g(motion)1567 1659 y(_)-38 b Fz(q)26 b FA(=)c(\()p Fz(\025)1782 1671 y Fv(1)1820 1659 y Fz(;)14 b(:)g(:)g(:)g(;)g(\025)2053 1671 y Fw(n)2098 1659 y FA(\))24 b(=:)e Fz(!)s(:)866 b FA(\(5.6\))515 1794 y(So)20 b(all)g(solutions)g(for)f(the)i(linear)f (equation)g(in)g Fz(T)2032 1764 y Fw(n)2077 1794 y FA(\()p Fz(I)7 b FA(\))21 b(are)e(quasip)r(erio)r(dic)h(curv)n(es)f(with)i(the) 515 1894 y(frequency-v)n(ector)27 b Fz(!)s FA(.)42 b(Our)29 b(goal)f(in)i(this)g(section)f(is)h(to)f(presen)n(t)g(and)g(discuss)g (a)g(KAM-)515 1994 y(theorem)35 b(whic)n(h)g(implies)h(that)f(under)h (certain)e(conditions)h(`most)h(of)6 b(')36 b(tra)5 b(jectories)33 b(of)515 2093 y(the)27 b(equation)g(\(5.6\))g(on)g(the)g(torus)g Fz(T)1725 2063 y Fw(n)1769 2093 y FA(\()p Fz(I)7 b FA(\))28 b(p)r(ersist)f(as)f(time-quasip)r(erio)r(dic)h(solutions)f(of)515 2193 y(the)i(p)r(erturb)r(ed)g(equation)f(\(5.1\))o(,)h(if)g Fz(")23 b(>)g FA(0)k(is)g(su\016cien)n(tly)h(small.)639 2292 y(T)-7 b(o)41 b(state)g(the)h(result)f(w)n(e)g(assume)f(that)i (the)g(op)r(erator)d Fz(A)j FA(and)f(the)g(function)h Fz(H)515 2392 y FA(analytically)27 b(dep)r(end)h(on)g(an)g(additional)f Fz(n)p FA(-dimensional)g(parameter)g Fz(a)c Fy(2)i(A)p FA(,)j(where)g Fy(A)515 2492 y FA(is)e(a)f(connected)h(b)r(ounded)g(op) r(en)g(domain)g(in)g Fx(R)2029 2462 y Fw(n)2080 2492 y FA(.)37 b(Then)26 b Fz(\025)2403 2504 y Fw(j)2461 2492 y FA(=)d Fz(\025)2597 2504 y Fw(j)2632 2492 y FA(\()p Fz(a)p FA(\).)37 b(W)-7 b(e)27 b(assume)e(that)515 2591 y(the)j(\014rst)f Fz(n)h FA(frequencies)f Fz(\025)1380 2603 y Fw(l)1429 2591 y FA(=)22 b Fz(!)1568 2603 y Fw(l)1621 2591 y FA(dep)r(end)28 b(on)g Fz(a)f FA(in)h(the)g(nondegenerate)e(w)n (a)n(y:)639 2732 y(H1\))i(det)p Fy(f)p Fz(@)5 b(!)1061 2744 y Fw(l)1086 2732 y Fz(=@)g(a)1221 2744 y Fw(k)1284 2732 y Fy(j)23 b FA(1)f Fy(\024)h Fz(k)s(;)28 b(l)c Fy(\024)f Fz(n)p Fy(g)f(6\021)h FA(0;)515 2872 y(and)k(that)h(the)g(follo)n(wing) f(sp)r(ectral)g(asymptotic)g(holds:)639 3032 y(H2\))803 2937 y Fq(\014)803 2986 y(\014)803 3036 y(\014)831 3032 y Fz(\025)879 3044 y Fw(j)914 3032 y FA(\()p Fz(a)p FA(\))19 b Fy(\000)f Fz(K)1195 3044 y Fv(1)1232 3032 y Fz(j)1271 3002 y Fw(d)1306 3010 y Ft(1)1360 3032 y Fy(\000)g Fz(K)1520 3002 y Fv(1)1514 3053 y(1)1557 3032 y Fz(j)1596 3002 y Fw(d)1631 2977 y Ft(1)1631 3019 y(1)1686 3032 y Fy(\000)g Fz(K)1846 3002 y Fv(2)1840 3053 y(1)1882 3032 y Fz(j)1921 3002 y Fw(d)1956 2977 y Ft(2)1956 3019 y(1)2011 3032 y Fy(\000)g Fz(:)c(:)g(:)2204 2937 y Fq(\014)2204 2986 y(\014)2204 3036 y(\014)2255 3032 y Fy(\024)23 b Fz(K)6 b(j)2470 2987 y Fv(~)2459 3002 y Fw(d)2497 3032 y Fz(;)65 b FA(Lip)13 b Fz(\025)2767 3044 y Fw(j)2826 3032 y Fy(\024)23 b Fz(j)2964 2987 y Fv(~)2953 3002 y Fw(d)2991 3032 y FA(,)515 3172 y(where)31 b Fz(d)802 3184 y Fv(1)869 3172 y FA(:=)f Fz(d)1030 3184 y Fw(A)1106 3172 y FA(+)21 b Fz(d)1235 3184 y Fw(J)1311 3172 y Fy(\025)30 b FA(1,)i Fz(K)1574 3184 y Fv(1)1641 3172 y Fz(>)e FA(0,)1848 3150 y(~)1833 3172 y Fz(d)g(<)g(d)2044 3184 y Fv(1)2103 3172 y Fy(\000)21 b FA(1)31 b(and)h(the)g(dots)f(stand)h(for)f(a)h(\014nite) 515 3271 y(sum)27 b(with)i(exp)r(onen)n(ts)e Fz(d)1313 3283 y Fv(1)1373 3271 y Fz(>)c(d)1504 3241 y Fv(1)1504 3292 y(1)1564 3271 y Fz(>)g(d)1695 3241 y Fv(2)1695 3292 y(1)1756 3271 y Fz(>)f(:)14 b(:)g(:)g FA(.)639 3412 y(Let)24 b(us)g(denote)f(b)n(y)h Fz(X)1338 3381 y Fw(c)1331 3432 y(s)1395 3412 y FA(the)g(complexi\014cation)e(of)i(a)f(space)g Fz(X)2588 3424 y Fw(s)2647 3412 y FA(and)h(assume)f(that)h(the)515 3511 y(equation)j(\(5.1\))g(is)h(quasilinear)e(and)h(analytic:)639 3651 y(H3\))i(the)f(set)h Fz(X)1147 3663 y Fw(d)1182 3671 y Ft(0)1237 3651 y Fy(\002)18 b(A)29 b FA(admits)f(in)h Fz(X)1862 3621 y Fw(c)1855 3675 y(d)1890 3683 y Ft(0)1944 3651 y Fy(\002)19 b Fx(C)2082 3621 y Fw(n)2161 3651 y FA(a)28 b(complex)g(neigh)n(b)r(ourho)r(o)r(d)f Fz(Q)h FA(suc)n(h)515 3751 y(that)c(the)h(map)49 b Fy(r)1106 3763 y Fw(x)1148 3751 y Fz(H)30 b FA(:)23 b Fz(Q)f Fy(!)i Fz(X)1564 3721 y Fw(c)1557 3774 y(d)1592 3782 y Ft(0)1623 3774 y Fu(\000)p Fw(d)1710 3782 y Fn(H)1816 3751 y FA(is)g (complex-analytic)f(and)h(b)r(ounded)h(uniformly)515 3868 y(on)i(b)r(ounded)h(subsets)f(of)h Fz(Q)p FA(.)37 b(Moreo)n(v)n(er,)24 b Fz(d)1907 3880 y Fw(H)1989 3868 y FA(+)18 b Fz(d)2115 3880 y Fw(J)2185 3868 y Fy(\024)2287 3847 y FA(~)2272 3868 y Fz(d)q FA(.)639 4009 y(Finally)-7 b(,)28 b(w)n(e)f(shall)g(need)h(the)g(follo)n(wing)f(non-resonance)e (condition:)639 4149 y(H4\))31 b(F)-7 b(or)30 b(all)h(in)n(teger)e Fz(n)p FA(-v)n(ectors)g Fz(s)i FA(and)f(\()p Fz(M)2064 4161 y Fv(2)2122 4149 y Fy(\000)20 b Fz(n)p FA(\)-v)n(ectors)29 b Fz(l)j FA(suc)n(h)e(that)h Fy(j)p Fz(s)p Fy(j)d(\024)g Fz(M)3319 4161 y Fv(1)3356 4149 y FA(,)515 4248 y(1)22 b Fy(\024)h(j)p Fz(l)r Fy(j)g(\024)f FA(2)27 b(w)n(e)h(ha)n(v)n(e,)1117 4408 y Fz(s)18 b Fy(\001)h Fz(!)s FA(\()p Fz(a)p FA(\))f(+)g Fz(l)1505 4420 y Fw(n)p Fv(+1)1634 4408 y Fz(\025)1682 4420 y Fw(n)p Fv(+1)1812 4408 y FA(\()p Fz(a)p FA(\))h(+)f Fy(\001)c(\001)g(\001)k FA(+)g Fz(l)2245 4420 y Fw(M)2308 4428 y Ft(2)2345 4408 y Fz(\025)2393 4420 y Fw(M)2456 4428 y Ft(2)2493 4408 y FA(\()p Fz(a)p FA(\))24 b Fy(6\021)e FA(0)p Fz(;)431 b FA(\(5.7\))515 4567 y(where)27 b(the)h(in)n(tegers)e Fz(M)1286 4579 y Fv(1)1346 4567 y Fz(>)d FA(0)k(and)g Fz(M)1745 4579 y Fv(2)1805 4567 y Fz(>)c(n)k FA(are)g(to)g(b)r(e)h(sp)r (eci\014ed.)639 4667 y(Relations)g(\(5.7\))f(with)i Fy(j)p Fz(l)r Fy(j)23 b FA(=)g(1)28 b(and)f Fy(j)p Fz(l)r Fy(j)d FA(=)f(2)k(are)g(called,)h(resp)r(ectiv)n(ely)-7 b(,)27 b(the)i(\014rst)e(and)515 4766 y(the)h(second)f Fp(Melnikov)k(c)l (ondition)6 b FA(.)639 4907 y(Let)34 b(us)g(\014x)f(an)n(y)g Fz(I)1229 4919 y Fv(0)1300 4907 y Fy(2)g Fx(R)1442 4876 y Fw(n)1442 4927 y Fv(+)1537 4907 y FA(and)h(denote)f(b)n(y)h(\006)2160 4919 y Fv(0)2230 4907 y FA(the)g(map)g Fx(T)2626 4876 y Fw(n)2693 4907 y Fy(\002)22 b(A)33 b(!)h Fz(X)3065 4919 y Fw(d)3100 4927 y Ft(0)3169 4907 y FA(whic)n(h)515 5006 y(sends)27 b(\()p Fz(q)s(;)14 b(a)p FA(\))28 b(to)g(the)g(p)r(oin) n(t)f(of)h(the)g(torus)f Fz(T)1923 4976 y Fw(n)1967 5006 y FA(\()p Fz(I)2035 5018 y Fv(0)2073 5006 y FA(\))h(with)g(the)g(co)r (ordinate)e Fz(q)s FA(.)1905 5255 y(14)p eop PStoPSsaved restore %%Page: (14,15) 8 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 15 14 bop 515 523 a FB(Theorem)31 b(5.1.)41 b Fp(Supp)l(ose)31 b(the)g(assumptions)g(H1\)-H3\))g(hold.)43 b(Then)31 b(ther)l(e)g(exist)f(inte-)515 623 y(gers)i Fz(M)773 635 y Fv(1)836 623 y Fz(>)26 b FA(0)31 b Fp(and)h Fz(M)1244 635 y Fv(2)1307 623 y Fz(>)26 b(n)31 b Fp(such)h(that)g(if)g(H4\))g(is) g(ful\014l)t(le)l(d,)h(then)e(for)i(arbitr)l(ary)g Fz(\015)e(>)25 b FA(0)515 722 y Fp(and)39 b(for)h(su\016ciently)g(smal)t(l)g Fz(")f(<)45 b FA(\026)-47 b Fz(")p FA(\()p Fz(\015)5 b FA(\))p Fp(,)42 b(a)d(Bor)l(el)h(subset)f Fy(A)2520 734 y Fw(")2595 722 y Fy(\032)h(A)f Fp(and)h(a)f(Lipschitz)515 822 y(map)32 b FA(\006)759 834 y Fw(")822 822 y FA(:)27 b Fx(T)928 792 y Fw(n)992 822 y Fy(\002)20 b(A)1143 834 y Fw(")1206 822 y Fy(!)27 b Fz(X)1385 834 y Fw(d)1420 842 y Ft(0)1456 822 y Fp(,)33 b(analytic)g(in)f Fz(q)e Fy(2)d Fx(T)2139 792 y Fw(n)2184 822 y Fp(,)33 b(c)l(an)e(b)l(e)h (found)h(with)f(the)h(fol)t(lowing)515 922 y(pr)l(op)l(erties:)639 1021 y(a\))d(mes)44 b FA(\()p Fy(AnA)1135 1033 y Fw(")1171 1021 y FA(\))23 b Fy(\024)g Fz(\015)5 b Fp(;)639 1121 y(b\))30 b(the)g(map)g FA(\006)1121 1133 y Fw(")1187 1121 y Fp(is)g Fz(C)6 b(")p Fp(-close)30 b(to)g FA(\006)1773 1133 y Fv(0)1810 1121 y Fy(j)1833 1133 y Fo(T)1878 1117 y Fn(n)1913 1133 y Fu(\002A)2018 1141 y Fn(")2084 1121 y Fp(in)g(the)g(Lipschitz)h(norm;)639 1220 y(c\))36 b(e)l(ach)h(torus)e FA(\006)1219 1232 y Fw(")1254 1220 y FA(\()p Fx(T)1342 1190 y Fw(n)1410 1220 y Fy(\002)22 b(f)p Fz(a)p Fy(g)p FA(\))p Fp(,)37 b Fz(a)c Fy(2)h(A)1951 1232 y Fw(")1987 1220 y Fp(,)k(is)e(invariant)g(for)h(the)e(\015ow)h(of)h(e)l(quation) 515 1320 y FA(\(5.1\))g Fp(and)i(is)f(\014l)t(le)l(d)h(with)g(its)f (time-quasip)l(erio)l(dic)i(solutions)f(of)g(the)f(form)h Fz(u)3070 1332 y Fw(")3105 1320 y FA(\()p Fz(t)p FA(;)14 b Fz(q)s FA(\))38 b(=)515 1420 y(\006)575 1432 y Fw(")610 1420 y FA(\()p Fz(q)23 b FA(+)d Fz(!)842 1390 y Fu(0)865 1420 y Fz(t;)14 b(a)p FA(\))p Fp(,)32 b Fz(q)e Fy(2)c Fx(T)1269 1390 y Fw(n)1314 1420 y Fp(,)33 b(wher)l(e)f(the)g(fr)l(e)l (quency)f(ve)l(ctor)h Fz(!)2417 1390 y Fu(0)2440 1420 y FA(\()p Fz(a)p FA(\))g Fp(is)g Fz(C)6 b(")p Fp(-close)33 b(to)e Fz(!)s FA(\()p Fz(a)p FA(\))h Fp(in)515 1519 y(the)e(Lipschitz)h (norm;)639 1619 y(d\))f(the)g(solutions)g Fz(u)1279 1631 y Fw(")1344 1619 y Fp(ar)l(e)g(line)l(arly)h(stable.)2046 1589 y Fv(6)639 1785 y FA(If)c Fy(r)p Fz(H)34 b FA(de\014nes)27 b(an)g(analytic)f(map)g(of)h(order)e Fz(d)2131 1797 y Fw(H)2222 1785 y FA(on)h(ev)n(ery)g(space)g Fz(X)2845 1797 y Fw(d)2883 1785 y FA(,)h Fz(d)c Fy(\025)g Fz(d)3130 1797 y Fv(0)3167 1785 y FA(,)28 b(then)515 1885 y(the)i(solutions)g Fz(u)1058 1897 y Fw(")1093 1885 y FA(,)h(constructed)f(in)h(the)f (theorem,)h(are)e(smo)r(oth.)45 b(Indeed,)31 b(if)g Fz(u)3091 1897 y Fw(")3126 1885 y FA(\()p Fz(t)p FA(\))g(is)f(a)515 1984 y(solution,)35 b(then)g(due)g(to)f(the)h(equation)e Fz(J)8 b(Au)1987 1996 y Fw(")2023 1984 y FA(\()p Fz(t)p FA(\))35 b(is)f(a)g(smo)r(oth)g(curv)n(e)f(in)i Fz(X)3019 1996 y Fw(d)3054 2004 y Ft(0)3085 1996 y Fu(\000)p Fw(d)3172 2004 y Fn(H)3225 1996 y Fu(\000)p Fw(d)3312 2004 y Fn(J)3356 1984 y FA(.)515 2084 y(Since)d Fz(J)8 b(A)32 b FA(is)f(an)h (automorphism)f(of)g(the)h(scale)f(of)h(order)e Fz(d)2456 2096 y Fv(1)2494 2084 y FA(,)j(then)f Fz(u)2791 2096 y Fw(")2826 2084 y FA(\()p Fz(t)p FA(\))g(is)g(a)f(smo)r(oth)515 2183 y(curv)n(e)e(in)i Fz(X)908 2195 y Fw(d)943 2203 y Ft(0)974 2195 y Fu(\000)p Fw(d)1061 2203 y Fn(H)1114 2195 y Fu(\000)p Fw(d)1201 2203 y Fn(J)1241 2195 y Fv(+)p Fw(d)1327 2203 y Ft(1)1391 2183 y Fy(\032)c Fz(X)1552 2195 y Fw(d)1587 2203 y Ft(0)1619 2195 y Fv(+1)1707 2183 y FA(.)45 b(Iterating)30 b(this)g(argumen)n(ts)f(w)n(e)h(see)g(that)h Fz(u)3186 2195 y Fw(")3251 2183 y FA(is)f(a)515 2283 y(smo)r(oth)d(curv)n(e)g(in)h(eac)n(h)e(space)h Fz(X)1604 2295 y Fw(s)1639 2283 y FA(.)639 2383 y(In)f(the)g(semilinear)e(case)h (\(i.e.,)h(when)g Fz(d)1891 2395 y Fw(H)1968 2383 y FA(+)14 b Fz(d)2090 2395 y Fw(J)2159 2383 y Fy(\024)2262 2361 y FA(~)2247 2383 y Fz(d)23 b(<)g(d)2444 2395 y Fv(1)2496 2383 y Fy(\000)13 b FA(1)25 b(and)2815 2361 y(~)2800 2383 y Fz(d)f Fy(\024)e FA(0\))k(the)g(theo-)515 2482 y(rem)h(is)h(pro)n(v)n(ed)e(in)i([Kuk87)n(,)g(Kuk88)n(])g(\(see)g(also) e([Kuk93)o(,)i(P\177)-42 b(os96a)m(]\).)37 b(The)28 b(semilinearit)n(y) 515 2582 y(restriction)920 2560 y(~)906 2582 y Fz(d)23 b Fy(\024)g FA(0)d(w)n(as)g(remo)n(v)n(ed)g(in)h([Kuk98)o(])g(\(see)g (also)f(\([Kuk00)o(])h(and)g([KP03)n(]\).)35 b(Sim)n(ul-)515 2682 y(taneously)22 b(with)i([Kuk87)o(,)f(Kuk88)o(])g(a)g(related)g (KAM-theorem)f(for)h(in\014nite-dimensional)515 2781 y(Hamiltonian)38 b(systems)g(with)h(short)f(in)n(teractions)g(w)n(as)f (pro)n(v)n(ed)h(b)n(y)g(P\177)-42 b(osc)n(hel)37 b([P\177)-42 b(os89)n(].)515 2881 y(The)32 b(systems)f(\(5.1\),)i(de\014ned)f(b)n(y) g(HPDEs,)g(are)f(not)h(short-in)n(terected,)f(but)i(results)e(of)515 2980 y([P\177)-42 b(os89)n(])32 b(apply)h(to)f(some)g(equations)g(from) g(non-equilibrium)g(statistical)g(ph)n(ysics.)52 b(W)-7 b(e)515 3080 y(note)19 b(that)g(for)g(the)h(short-in)n(terected)e (systems)h(a)f(KAM-theory)g(for)h(in\014nite-dimensional)515 3180 y(in)n(v)-5 b(arian)n(t)26 b(tori)h(also)g(is)g(a)n(v)-5 b(ailable,)27 b(see)g([W)-7 b(a)n(y84)n(,)28 b(P\177)-42 b(os90)n(])28 b(and)f(references)f(in)i([P\177)-42 b(os90)n(].)639 3279 y(F)-7 b(or)26 b(some)g(sp)r(eci\014c)h(HPDEs)f(\(5.1\))g(the)h (assertions)e(of)h(Theorem)g(5.1)g(can)g(b)r(e)h(pro)n(v)n(en)515 3379 y(for)32 b(an)n(y)f Fz(n)g Fy(\025)g FA(1)h(ev)n(en)g(if)h(the)g (parameter)e Fz(a)h FA(is)h(only)f(one-dimensional.)50 b(In)32 b(particular,)515 3479 y(this)21 b(can)g(b)r(e)h(done)f(for)g (the)h(nonlinear)e(w)n(a)n(v)n(e)g(equation)h(as)g(in)g(Example)g(5.3)f (b)r(elo)n(w,)j(where)515 3578 y Fz(V)c FA(\()p Fz(x)p FA(\))35 b Fy(\021)f Fz(a)g FA(and)g(the)h(constan)n(t)f Fz(a)g FA(is)g(the)h(one-dimensional)e(parameter.)56 b(See)34 b([Bou94)o(])515 3678 y(and)27 b([Bam99b)o(].)639 3778 y(The)35 b(pro)r(of)e(of)h(Theorem)g(5.1)f(is)h(rather)g(tec)n (hnical.)56 b(F)-7 b(or)34 b(its)g(w)n(ell-written)g(outline)515 3877 y(in)29 b(the)h(semilinear)e(case)g(see)h([Cra00)n(].)42 b(Belo)n(w)28 b(w)n(e)h(presen)n(t)f(the)i(pro)r(of)6 b('s)29 b(sc)n(heme)f(in)i(the)515 3977 y(form)d(whic)n(h)h(suits)f (further)h(discussions.)515 4126 y FB(The)38 b(sc)m(heme)e(of)h(the)h (pro)s(of)f(of)g(Theorem)f(5.1.)51 b FA(W)-7 b(e)33 b(start)f(with)h (the)g(semilinear)515 4226 y(case)h(and)h(assume)g(for)g(simplicit)n(y) g(that)g Fz(\025)1918 4196 y Fw(J)1918 4247 y(j)2001 4226 y Fy(\021)g FA(1.)60 b(Then)35 b Fz(I)43 b FA(=)36 b(\()p Fz(I)2698 4238 y Fv(1)2736 4226 y Fz(;)14 b(:)g(:)g(:)f(;)h(I) 2956 4238 y Fw(n)3002 4226 y FA(\))35 b(and)h Fz(q)i FA(=)515 4325 y(\()p Fz(q)584 4337 y Fv(1)621 4325 y Fz(;)14 b(:)g(:)g(:)g(;)g(q)843 4337 y Fw(n)888 4325 y FA(\))38 b(form)g(a)g(symplectic)g(co)r(ordinate)f(system)h(in)g(the) g(space)g(\(2.3\).)68 b(W)-7 b(e)38 b(set)515 4425 y Fz(Y)52 b FA(=)33 b(span)p Fy(f)p Fz(')976 4390 y Fu(\006)976 4448 y Fw(j)1032 4425 y Fz(;)27 b(j)39 b(>)33 b(n)p Fy(g)g(\032)h Fz(X)7 b FA(,)35 b(and)f(denote)g(b)n(y)f Fz(y)2218 4390 y Fu(\006)2215 4448 y Fw(j)2274 4425 y FA(,)j Fz(j)i(>)c(n)p FA(,)h(the)g(co)r(ordinates)d(in)i Fz(Y)515 4539 y FA(with)i(resp)r (ect)f(to)h(the)g(basis)f Fy(f)p Fz(')1573 4504 y Fu(\006)1573 4562 y Fw(j)1629 4539 y Fy(g)p FA(.)61 b(T)-7 b(o)35 b(study)h(the)g(vicinit)n(y)g(of)g(a)f(torus)g Fz(T)3041 4509 y Fw(n)3085 4539 y FA(\()p Fz(I)3153 4551 y Fv(0)3191 4539 y FA(\),)j(w)n(e)515 4650 y(mak)n(e)28 b(the)i(substitution)f Fz(I)k FA(=)25 b Fz(I)1536 4662 y Fv(0)1593 4650 y FA(+)19 b Fz(p)p FA(.)42 b(Then)2022 4629 y(\026)2002 4650 y Fz(J)8 b(du)19 b Fy(^)h Fz(du)25 b FA(=)g Fz(dp)20 b Fy(^)g Fz(dq)i FA(+)d Fz(dy)2900 4620 y Fv(+)2975 4650 y Fy(^)h Fz(dy)3137 4620 y Fu(\000)3193 4650 y FA(,)29 b(and)p 515 4703 1146 4 v 607 4757 a Fm(6)642 4780 y Fl(If)22 b(equation)i(\(5.1\))f(is)f(not)h(semilinear)d(\(i.e.,)i(if)f Fk(d)1950 4791 y Fg(J)2006 4780 y Fl(+)13 b Fk(d)2110 4791 y Fg(H)2187 4780 y Fk(>)20 b Fl(0\),)j(then)g(this)f(assertion)h (is)e(pro)n(v)n(ed)i(pro-)515 4859 y(vided)g(that)i(the)f(equation)h (satis\014es)e(some)g(mild)e(regularit)n(y)i(condition,)h(see)g (Theorem)e(8.4)h(in)g([Kuk00)q(].)1905 5255 y FA(15)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 16 15 bop 515 523 a Fz(T)576 493 y Fw(n)620 523 y FA(\()p Fz(I)688 535 y Fv(0)726 523 y FA(\))23 b(=)g Fy(f)p Fz(p)f FA(=)h(0)p Fz(;)k(y)f FA(=)d(0)p Fy(g)p FA(.)36 b(In)27 b(the)h(new)g(v)-5 b(ariables)26 b(equation)h(\(2.1\))h(tak)n(es)e(the) i(form)1223 706 y(_)-38 b Fz(q)26 b FA(=)c Fy(r)1427 718 y Fw(p)1466 706 y Fy(H)1536 718 y Fw(")1572 706 y Fz(;)113 b FA(_)-40 b Fz(p)23 b FA(=)g Fy(\000r)1978 718 y Fw(q)2014 706 y Fy(H)2084 718 y Fw(")2120 706 y Fz(;)112 b FA(_)-38 b Fz(y)26 b FA(=)c Fz(J)8 b Fy(r)2517 718 y Fw(y)2557 706 y Fy(H)2627 718 y Fw(")2663 706 y Fz(;)515 888 y FA(with)28 b(the)g(hamiltonian)976 1071 y Fy(H)1046 1083 y Fw(")1105 1071 y FA(=)22 b Fz(H)1261 1083 y Fv(0)1299 1071 y FA(\()p Fz(p;)14 b(y)s FA(\))k(+)g Fz("H)1695 1083 y Fv(1)1732 1071 y FA(\()p Fz(p;)c(q)s(;)g(y)s FA(\))p Fz(;)97 b(H)2185 1083 y Fv(0)2245 1071 y FA(=)23 b Fz(!)e Fy(\001)d Fz(p)h FA(+)2601 1038 y Fv(1)p 2601 1052 34 4 v 2601 1100 a(2)2644 1071 y Fy(h)p Fz(Ay)s(;)14 b(y)s Fy(i)p Fz(:)290 b FA(\(5.8\))515 1254 y(The)29 b(v)n(ector)e Fz(!)32 b FA(and)d(the)g(op)r(erator)f Fz(A)h FA(dep)r(end)h(on)e(the)i(parameter)d Fz(a)p FA(;)j(the)f (function)h Fz(H)3342 1266 y Fv(1)515 1353 y FA(dep)r(ends)e(on)f Fz(a)h FA(and)f Fz(I)1220 1365 y Fv(0)1258 1353 y FA(.)37 b(W)-7 b(e)28 b(call)f Fz(H)1682 1365 y Fv(0)1747 1353 y FA(the)h(in)n(tegrable)e(part)h(of)h(the)g(hamiltonian)f Fy(H)3224 1365 y Fw(")3260 1353 y FA(.)639 1453 y(Retaining)e(the)h (terms)e(of)h Fz(H)1546 1465 y Fv(1)1609 1453 y FA(whic)n(h)g(are)f (a\016ne)h(in)g Fz(p)g FA(and)g(quadratic)f(in)h Fz(y)s FA(,)h(w)n(e)e(write)515 1553 y Fz(H)584 1565 y Fv(1)649 1553 y FA(as)649 1735 y Fz(H)718 1747 y Fv(1)779 1735 y FA(=)e Fz(H)942 1701 y Fv(1)935 1756 y(1)998 1735 y FA(+)c Fz(H)1157 1701 y Fv(3)1150 1756 y(1)1194 1735 y Fz(;)97 b(H)1390 1701 y Fv(1)1383 1756 y(1)1450 1735 y FA(=)22 b Fz(h)p FA(\()p Fz(q)s FA(\))d(+)f Fz(h)1839 1701 y Fw(p)1878 1735 y FA(\()p Fz(q)s FA(\))h Fy(\001)f Fz(p)g FA(+)g Fy(h)p Fz(h)2265 1701 y Fw(y)2305 1735 y FA(\()p Fz(q)s FA(\))p Fz(;)c(y)s Fy(i)19 b FA(+)f Fy(h)p Fz(h)2704 1701 y Fw(y)r(y)2780 1735 y FA(\()p Fz(q)s FA(\))p Fz(y)s(;)c(y)s Fy(i)p Fz(;)1515 1860 y(H)1591 1825 y Fv(3)1584 1880 y(1)1651 1860 y FA(=)22 b Fz(O)r FA(\()p Fy(j)p Fz(p)p Fy(j)1923 1825 y Fv(2)1980 1860 y FA(+)c Fy(k)p Fz(y)s Fy(k)2191 1825 y Fv(3)2245 1860 y FA(+)g Fy(j)p Fz(p)p Fy(j)c(k)p Fz(y)s Fy(k)p FA(\))22 b(=:)h Fy(O)r FA(\()p Fz(p;)14 b(q)s(;)g(y)s FA(\))p Fz(:)639 2042 y FA(Next)28 b(in)f(the)g(vicinit)n(y)g(of)g(the)g(torus) f Fz(T)1886 2012 y Fw(n)1954 2042 y FA(=)c Fy(f)p Fz(p)h FA(=)f(0)p Fz(;)27 b(y)f FA(=)d(0)p Fy(g)j FA(w)n(e)g(mak)n(e)g(a)h (symplectic)515 2142 y(c)n(hange)k(of)h(v)-5 b(ariable)32 b(to)g(kill)h(the)g(part)f Fz("H)1910 2112 y Fv(1)1903 2163 y(1)1979 2142 y FA(of)g(the)h(p)r(erturbation)f Fz("H)2826 2154 y Fv(1)2863 2142 y FA(.)52 b(This)32 b(c)n(hange)515 2242 y(of)27 b(v)-5 b(ariable)26 b(is)g(a)h (transformation)e Fz(S)1687 2254 y Fv(1)1752 2242 y FA(whic)n(h)h(is)h (the)h(time-)p Fz(")f FA(shift)g(along)f(tra)5 b(jectories)25 b(of)515 2341 y(an)h(additional)g(hamiltonian)g Fz(F)12 b FA(.)36 b(Here)27 b(the)f(recip)r(e)h(is)f(that)h(to)f(kill)h Fz(H)2760 2311 y Fv(1)2753 2362 y(1)2797 2341 y FA(,)g Fz(F)38 b FA(should)26 b(b)r(e)h(of)515 2441 y(the)j(same)f(structure,) h(so)f Fz(F)39 b FA(=)26 b Fz(f)9 b FA(\()p Fz(q)s FA(\))21 b(+)e Fz(f)1848 2411 y Fw(p)1886 2441 y FA(\()p Fz(q)s FA(\))i Fy(\001)f Fz(p)f FA(+)h Fy(h)p Fz(f)2282 2411 y Fw(y)2322 2441 y FA(\()p Fz(q)s FA(\))p Fz(;)14 b(y)s Fy(i)20 b FA(+)f Fy(h)p Fz(f)2725 2411 y Fw(y)r(y)2801 2441 y FA(\()p Fz(q)s FA(\))p Fz(y)s(;)14 b(y)s Fy(i)p FA(.)44 b(Due)30 b(to)515 2540 y(Theorem)d(3.4)f(w)n(e)h(can)h(write)f (the)h(transformed)f(hamiltonian)g Fy(H)2625 2552 y Fw(")2679 2540 y Fy(\016)18 b Fz(S)2790 2552 y Fv(1)2855 2540 y FA(as)515 2723 y Fy(H)585 2735 y Fw(")626 2723 y Fy(\016)6 b Fz(S)725 2735 y Fv(1)784 2723 y FA(=)23 b Fz(H)941 2735 y Fv(0)984 2723 y FA(+)6 b Fz("H)1163 2735 y Fv(1)1205 2723 y FA(+)g Fz(")p Fy(h)p Fz(J)i Fy(r)1470 2735 y Fw(y)1509 2723 y Fz(F)r(;)28 b Fy(r)1684 2735 y Fw(y)1724 2723 y Fz(H)1793 2735 y Fv(0)1831 2723 y Fy(i)6 b FA(+)g Fz(")p Fy(r)2048 2735 y Fw(p)2085 2723 y Fz(F)18 b Fy(\001)6 b(r)2254 2735 y Fw(q)2290 2723 y Fz(H)2359 2735 y Fv(0)2402 2723 y Fy(\000)g Fz(")p Fy(r)2581 2735 y Fw(q)2617 2723 y Fz(F)17 b Fy(\001)6 b(r)2785 2735 y Fw(p)2824 2723 y Fz(H)2893 2735 y Fv(0)2935 2723 y FA(+)g Fz(O)r FA(\()p Fz(")3142 2689 y Fv(2)3179 2723 y FA(\))g(+)g Fy(O)r Fz(:)515 2906 y FA(Since)24 b Fy(r)797 2918 y Fw(p)835 2906 y Fz(H)904 2918 y Fv(0)964 2906 y FA(=)f Fz(!)s FA(,)h Fy(r)1223 2918 y Fw(q)1260 2906 y Fz(H)1329 2918 y Fv(0)1390 2906 y FA(=)e(0)h(and)h Fy(r)1769 2918 y Fw(y)1809 2906 y Fz(H)1878 2918 y Fv(0)1938 2906 y FA(=)f Fz(Ay)s FA(,)h(then)h(the)f(linear)f(in)h Fz(")f FA(term)h(v)-5 b(anishes)515 3005 y(if)28 b(the)g(follo)n(wing)e(relations)h(hold:) 1734 3188 y(\()p Fz(!)22 b Fy(\001)c(r)p FA(\))p Fz(f)32 b FA(=)23 b Fz(h;)97 b FA(\()p Fz(!)21 b Fy(\001)e(r)p FA(\))p Fz(f)2609 3154 y Fw(p)2670 3188 y FA(=)k Fz(h)2806 3154 y Fw(p)2844 3188 y Fz(;)847 3313 y FA(\()p Fz(!)e Fy(\001)e(r)p FA(\))p Fz(f)1145 3278 y Fw(y)1203 3313 y Fy(\000)f Fz(J)8 b(Af)1452 3278 y Fw(y)1515 3313 y FA(=)23 b Fz(h)1651 3278 y Fw(y)1690 3313 y Fz(;)97 b FA(\()p Fz(!)22 b Fy(\001)c(r)p FA(\))p Fz(f)2108 3278 y Fw(y)r(y)2202 3313 y FA(+)g([)p Fz(f)2358 3278 y Fw(y)r(y)2434 3313 y Fz(;)27 b(J)8 b(A)p FA(])24 b(=)e Fz(h)2782 3278 y Fw(y)r(y)2858 3313 y Fz(:)515 3495 y FA(W)-7 b(e)19 b(tak)n(e)g(these)g(relations)f(as)g(equations)g(on)h Fz(f)9 b FA(,)21 b Fz(f)2061 3465 y Fw(p)2099 3495 y FA(,)g Fz(f)2193 3465 y Fw(y)2251 3495 y FA(and)e Fz(f)2454 3465 y Fw(y)r(y)2549 3495 y FA(\(called)g Fp(`the)j(homolo)l(gic)l(al) 515 3595 y(e)l(quations')10 b FA(\))29 b(and)e(try)g(to)h(solv)n(e)e (them.)639 3694 y(Since)i(the)f(equations)g(are)f(constan)n(t-co)r (e\016cien)n(t,)g(then)i(decomp)r(osing)e Fz(f)9 b FA(,)27 b Fz(f)3083 3664 y Fw(p)3121 3694 y Fz(;)h(:)14 b(:)g(:)41 b FA(in)515 3794 y(F)-7 b(ourier)32 b(series)h(in)h Fz(q)s FA(,)i(w)n(e)d(\014nd)h(for)f(their)h(comp)r(onen)n(ts)f(\(and)h(for)f (matrix)h(comp)r(onen)n(ts)515 3894 y(of)27 b(the)g(op)r(erator)e Fz(f)1135 3864 y Fw(y)r(y)1210 3894 y FA(\))i(explicit)h(form)n(ulae.) 35 b(Certain)26 b(terms)h(in)g(these)g(form)n(ulae)f(con)n(tain)515 3993 y(small)33 b(divisors,)h(whic)n(h)g(v)-5 b(anish)34 b(for)f(some)g(v)-5 b(alues)34 b(of)g(the)g(v)n(ector)f Fz(!)j FA(=)d Fz(!)s FA(\()p Fz(a)p FA(\).)56 b(Careful)515 4093 y(analysis)28 b(of)i(these)f(divisors)g(sho)n(w)f(that)i(all)g(of) f(them)h(are)f(b)r(ounded)h(a)n(w)n(a)n(y)e(from)h(zero)g(if)515 4193 y Fz(a)j(=)-51 b Fy(2)23 b(A)726 4205 y Fv(1)764 4193 y FA(,)j(where)g Fy(A)1118 4205 y Fv(1)1181 4193 y FA(is)g(a)g(Borel)f(subset)h(of)g Fy(A)h FA(of)f(small)f(measure.)36 b(When)26 b(the)h(equations)515 4292 y(are)39 b(solv)n(ed,)j(w)n(e)e (get)f(a)h(symplectic)g(transformation)f(whic)n(h)g(in)i(a)e (su\016cien)n(tly)h(small)515 4392 y(neigh)n(b)r(ourho)r(o)r(d)34 b(of)h Fz(T)1253 4362 y Fw(n)1332 4392 y FA(transforms)f(the)i (hamiltonian)f Fy(H)2444 4404 y Fw(")2515 4392 y FA(to)g(a)f (hamiltonian)h(whic)n(h)515 4491 y(di\013ers)27 b(from)g(its)h(in)n (tegrable)f(part)g(b)n(y)g Fz(O)r FA(\()p Fz(")1891 4461 y Fv(2)1929 4491 y FA(\).)639 4591 y(The)g(explanation)f(ab)r(o)n(v)n (e)g(has)g(some)h(\015o)n(ws.)36 b(The)27 b(most)f(imp)r(ortan)n(t)h (one)g(is)f(that)i(the)515 4691 y(\014rst)22 b(and)h(the)g(second)f (homological)e(equations)i(can)g(b)r(e)h(solv)n(ed)f(only)g(if)h(the)g (mean)f(v)-5 b(alues)515 4790 y(of)25 b Fz(h)g FA(and)g Fz(h)887 4760 y Fw(p)951 4790 y FA(v)-5 b(anish.)36 b(T)-7 b(o)25 b(ful\014ll)h(the)f(\014rst)g(condition)g(w)n(e)g(c)n(hange)f (the)i(hamiltonian)f Fz("H)3342 4802 y Fv(1)515 4890 y FA(b)n(y)i(a)g(constan)n(t)f(\(this)i(c)n(hange)f(is)g(irrelev)-5 b(an)n(t)26 b(since)h(it)h(do)r(es)f(not)g(a\013ect)h(the)g(equations)e (of)515 4990 y(motion\),)k(while)h(to)e(ful\014ll)i(the)g(second)e(w)n (e)h(subtract)f(from)h Fz("H)2572 5002 y Fv(1)2639 4990 y FA(the)g(a)n(v)n(erage)e Fz(")p Fy(h)p Fz(h)3204 4959 y Fw(p)3242 4990 y Fy(i)20 b(\001)g Fz(p)1905 5255 y FA(16)p eop PStoPSsaved restore %%Page: (16,17) 9 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 17 16 bop 515 523 a FA(and)22 b(add)g(it)g(to)g(the)h(in)n(tegrable)e (part)g Fz(H)1761 535 y Fv(0)1799 523 y FA(,)i(th)n(us)f(c)n(hanging)f (the)h(term)h Fz(!)11 b Fy(\001)d Fz(p)20 b FA(to)i Fz(!)3003 493 y Fv(2)3048 523 y Fy(\001)8 b Fz(p)p FA(,)23 b(where)515 623 y Fz(!)570 593 y Fv(2)630 623 y FA(=)f Fz(!)10 b FA(+)d Fz(")p Fy(h)p Fz(h)970 593 y Fw(p)1008 623 y Fy(i)p FA(.)36 b(Similar,)23 b(to)f(solv)n(e)e(the)j(last)f(homological)e (equation)h(w)n(e)h(subtract)f(from)515 722 y(the)36 b(op)r(erator)e Fz(h)1057 692 y Fw(y)r(y)1169 722 y FA(the)i(a)n(v)n (erage)d(of)j(its)g(diagonal)f(part)g(and)h(add)g(the)g(corresp)r (onding)515 822 y(quadratic)26 b(form)i(to)f Fz(H)1253 834 y Fv(0)1290 822 y FA(.)37 b(Th)n(us,)28 b(the)g(transformed)e (hamiltonian)h(b)r(ecomes)781 1005 y Fy(H)851 1017 y Fv(2)912 1005 y FA(:=)22 b Fy(H)1092 1017 y Fw(")1147 1005 y Fy(\016)c Fz(S)1258 1017 y Fv(1)1318 1005 y FA(=)k Fz(!)1457 1017 y Fv(2)1513 1005 y Fy(\001)c Fz(p)g FA(+)1707 972 y Fv(1)p 1707 986 34 4 v 1707 1033 a(2)1750 1005 y Fy(h)p Fz(A)1844 1017 y Fv(2)1882 1005 y Fz(y)s(;)28 b(y)s Fy(i)18 b FA(+)g Fz(")2193 970 y Fv(2)2230 1005 y Fz(H)2299 1017 y Fv(2)2336 1005 y FA(\()p Fz(p;)28 b(q)s(;)g(y)s FA(\))18 b(+)g Fy(O)r FA(\()p Fz(p;)29 b(q)s(;)e(y)s FA(\))p Fz(:)515 1187 y FA(This)g(transformation)f(is)i (called)f Fp(the)j(KAM-step)p FA(.)639 1287 y(Next)i(w)n(e)g(p)r (erform)f(the)h(second)f(KAM-step.)49 b(Under)32 b(the)g(condition)f (that)h Fz(a)39 b(=)-51 b Fy(2)30 b(A)3341 1299 y Fv(2)515 1386 y FA(w)n(e)23 b(\014nd)i(a)e(transformation)f Fz(S)1474 1398 y Fv(2)1536 1386 y FA(whic)n(h)h(sends)h(the)g(hamiltonian)g Fy(H)2655 1398 y Fv(2)2716 1386 y FA(to)g Fy(H)2884 1398 y Fv(3)2945 1386 y FA(=)e Fy(H)3102 1398 y Fv(2)3151 1386 y Fy(\016)11 b Fz(S)3255 1398 y Fv(2)3314 1386 y FA(=)515 1486 y Fz(!)567 1498 y Fv(3)623 1486 y Fy(\001)20 b Fz(p)f FA(+)822 1453 y Fv(1)p 822 1467 V 822 1515 a(2)865 1486 y Fy(h)p Fz(A)959 1498 y Fv(3)997 1486 y Fz(y)s(;)27 b(y)s Fy(i)20 b FA(+)f(\()p Fz(")1342 1456 y Fv(2)1379 1486 y FA(\))1411 1456 y Fv(2)1449 1486 y Fz(H)1518 1498 y Fv(2)1575 1486 y FA(+)g Fy(O)r FA(\()p Fz(p;)28 b(q)s(;)g(y)s FA(\),)i(etc.)42 b(After)30 b Fz(m)g FA(steps)f(w)n(e)g(\014nd)h (transfor-)515 1586 y(mations)d Fz(S)880 1598 y Fv(1)917 1586 y Fz(;)14 b(:)g(:)g(:)g(;)g(S)1153 1598 y Fw(m)1243 1586 y FA(suc)n(h)27 b(that)664 1768 y Fy(H)734 1780 y Fw(")788 1768 y Fy(\016)18 b Fz(S)899 1780 y Fv(1)955 1768 y Fy(\016)g(\001)c(\001)g(\001)k(\016)g Fz(S)1241 1780 y Fw(m)1327 1768 y FA(=)23 b Fz(!)1467 1780 y Fw(m)1548 1768 y Fy(\001)18 b Fz(p)h FA(+)1743 1736 y Fv(1)p 1743 1750 V 1743 1797 a(2)1786 1768 y Fy(h)p Fz(A)1880 1780 y Fw(m)1943 1768 y Fz(y)s(;)28 b(y)s Fy(i)18 b FA(+)g Fz(")2254 1734 y Fv(2)2287 1709 y Fn(m)2346 1768 y Fz(H)2415 1780 y Fw(m)2497 1768 y FA(+)g Fy(O)r FA(\()p Fz(p;)28 b(q)s(;)g(y)s FA(\))23 b(=:)f Fy(H)3143 1780 y Fw(m)3207 1768 y Fz(:)515 1951 y FA(The)31 b(torus)f Fz(T)966 1921 y Fw(n)1039 1951 y FA(=)e Fy(f)p Fz(p)g FA(=)h(0)p Fz(;)14 b(y)30 b FA(=)f(0)p Fy(g)h FA(is)h(`almost)f(in)n(v)-5 b(arian)n(t')30 b(for)g(the)i(equation)e(with)i(the)515 2051 y(hamiltonian)h Fy(H)1052 2063 y Fw(m)1115 2051 y FA(.)56 b(Hence,)35 b(the)f(torus)f Fz(S)1890 2063 y Fv(1)1950 2051 y Fy(\016)22 b(\001)14 b(\001)g(\001)22 b(\016)h Fz(S)2249 2063 y Fw(m)2311 2051 y FA(\()p Fz(T)2404 2021 y Fw(n)2449 2051 y FA(\))34 b(is)g(`almost)f(in)n(v)-5 b(arian)n(t')33 b(for)515 2150 y(the)c(original)f(one.)40 b(Since)29 b(the)h(sequence)e Fz(")1896 2120 y Fv(2)1929 2095 y Fn(m)2017 2150 y FA(con)n(v)n(erges)f(to)h(zero)g(sup)r(er-exp)r (onen)n(tially)515 2250 y(fast,)34 b(w)n(e)f(can)f(c)n(ho)r(ose)g(the)h (sets)g Fy(A)1640 2262 y Fv(1)1678 2250 y FA(,)h Fy(A)1801 2262 y Fv(2)1839 2250 y Fz(;)27 b(:)14 b(:)g(:)47 b FA(in)33 b(suc)n(h)g(a)g(w)n(a)n(y)e(that)j(mes\()p Fy(A)2999 2262 y Fu(1)3101 2250 y FA(=)e Fy(A)3264 2262 y Fv(1)3324 2250 y Fy([)515 2350 y(A)581 2362 y Fv(2)637 2350 y Fy([)19 b Fz(:)14 b(:)g(:)g FA(\))24 b Fz(<)f(\015)5 b FA(,)28 b(for)f(an)n(y)g Fz(a)33 b(=)-51 b Fy(2)24 b(A)1562 2362 y Fu(1)1660 2350 y FA(the)29 b(v)n(ectors)d Fz(!)2138 2362 y Fw(m)2201 2350 y FA(\()p Fz(a)p FA(\))i(con)n(v)n(erge)e(to)h(a) h(limiting)g(v)n(ector)515 2449 y Fz(!)570 2419 y Fu(0)593 2449 y FA(\()p Fz(a)p FA(\),)d(and)f(the)g(transformations)e Fz(S)1692 2461 y Fv(1)1741 2449 y Fy(\016)11 b(\001)j(\001)g(\001)d (\016)g Fz(S)2006 2461 y Fw(m)2093 2449 y FA(con)n(v)n(erge)21 b(to)j(a)g(limiting)g(map)g(\006)3138 2461 y Fw(")3174 2449 y FA(\()p Fy(\001)p Fz(;)k(a)p FA(\),)515 2549 y(de\014ned)d(on)f Fz(T)971 2519 y Fw(n)1015 2549 y FA(.)36 b(Then)25 b(the)g(torus)f (\006)1698 2561 y Fw(")1734 2549 y FA(\()p Fz(T)1827 2519 y Fw(n)1871 2549 y Fz(;)k(a)p FA(\))d(is)g(in)n(v)-5 b(arian)n(t)23 b(for)i(the)g(equation)f(\(5.1\))g(and)515 2648 y(is)j(\014lled)h(with)g(its)g(quasip)r(erio)r(dic)f(solutions)g Fz(t)c Fy(!)g FA(\006)2178 2660 y Fw(")2213 2648 y FA(\()p Fz(q)f FA(+)c Fz(!)2442 2618 y Fu(0)2465 2648 y Fz(t;)28 b(a)p FA(\).)639 2798 y(If)39 b(the)g(equation)f(is)h(not)g (semilinear,)h(then)f(the)g(situation)g(complicates)f(since)g(to)515 2897 y(solv)n(e)28 b(the)h(forth)h(homological)d(equation)i(w)n(e)g(ha) n(v)n(e)f(to)h(remo)n(v)n(e)e(from)i(the)h(op)r(erator)d Fz(h)3303 2867 y Fw(y)r(y)515 2997 y FA(the)32 b(whole)g(of)g(its)h (diagonal)d(part)i(\(not)g(only)g(its)g(a)n(v)n(erage\).)48 b(Because)32 b(of)g(that)g(the)h(op-)515 3097 y(erator)k Fz(A)i FA(in)h(the)f(in)n(tegrable)f(part)g(of)h(the)g(hamiltonian)g (gets)f(terms)h(whic)n(h)g(form)f(a)515 3196 y(small)26 b Fz(q)s FA(-dep)r(enden)n(t)h(diagonal)f(op)r(erator)f(of)h(a)h(p)r (ositiv)n(e)f(order.)35 b(Accordingly)-7 b(,)26 b(the)h(forth)515 3296 y(homological)32 b(equation)i(b)r(ecomes)g(more)g(di\016cult)h (and)f(cannot)g(b)r(e)h(solv)n(ed)e(b)n(y)h(the)h(di-)515 3396 y(rect)j(F)-7 b(ourier)37 b(metho)r(d.)69 b(Its)38 b(resolution)f(follo)n(ws)g(from)h(a)g(non-trivial)f(lemma,)k(based)515 3495 y(on)35 b(prop)r(erties)g(of)g(fast-oscillating)f(F)-7 b(ourier)35 b(in)n(tegrals,)h(pro)n(v)n(ed)e(in)i([Kuk98)o(])g(\(see)f (also)515 3595 y([Kuk00)n(,)28 b(KP03)n(]\).)515 3827 y Fs(5.2)112 b(Applications)35 b(to)i(1D)h(HPDEs)515 3981 y FA(Theorem)31 b(1)h(w)n(ell)g(applies)g(to)g(parameter-dep)r (ending)f(quasilinear)g(HPDEs)h(with)h(one-)515 4080 y(dimensional)28 b(space)f(v)-5 b(ariable)27 b(in)i(a)f(\014nite)h(in)n (terv)-5 b(al,)28 b(supplemen)n(ted)g(b)n(y)g(b)r(oundary)g(con-)515 4180 y(ditions)k(suc)n(h)f(that)i(sp)r(ectrum)f(of)g(the)g(linear)f(op) r(erator)f Fz(J)8 b(A)33 b FA(is)e(not)h(m)n(ultiple.)51 b(Indeed,)515 4279 y(for)23 b(suc)n(h)g(equations)f(assumption)h(H2\))g (follo)n(ws)g(from)g(usual)g(sp)r(ectral)f(asymptotics,)i(H3\))515 4379 y(is)g(ob)n(vious)f(if)i(the)g(nonlinearit)n(y)e(is)i(analytic,)f (while)h(H1\))g(and)f(H4\))g(hold)h(if)g(the)g(equation)515 4479 y(dep)r(ends)k(on)f(the)h(additional)f(parameter)f(in)i(a)g (non-degenerate)d(w)n(a)n(y)-7 b(.)39 b(More)28 b(explicitly)515 4578 y(it)d(means)f(the)h(follo)n(wing.)35 b(In)25 b(the)g(examples)f (whic)n(h)h(w)n(e)f(consider)g(b)r(elo)n(w,)h(the)g(equations)515 4678 y(dep)r(end)34 b(on)e(a)h(p)r(oten)n(tial)g Fz(V)19 b FA(\()p Fz(x)p FA(;)47 b Fz(a)p FA(\),)35 b(whic)n(h)e(is)g(analytic) g(in)g Fz(a)g FA(and)g(smo)r(oth)g(in)h Fz(x)p FA(.)54 b(The)515 4778 y(non-degeneracy)18 b(means)j(that)g(in)g(a)f (functional)h(space,)h(formed)e(b)n(y)h(functions)g(of)g Fz(x)g FA(and)g Fz(a)515 4877 y FA(of)k(the)h(required)e(smo)r (othness,)i(the)f(p)r(oten)n(tial)h Fz(V)44 b FA(should)25 b(not)h(b)r(elong)f(to)g(some)g(analytic)515 4977 y(subset)i(of)h (in\014nite)g(co)r(dimension.)1905 5255 y(17)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 18 17 bop 639 523 a FA(Belo)n(w)27 b(w)n(e)h(just)g(list)h(the)f (examples.)37 b(In)28 b(eac)n(h)g(case)f(application)g(of)h(Theorem)f (5.1)g(is)515 623 y(straigh)n(tforw)n(ard.)50 b(The)33 b(theorem)g(applies)g(if)g(dimension)g(of)g(the)h(parameter)d Fz(a)i FA(is)g Fy(\025)f Fz(n)515 722 y FA(and)g(dep)r(endence)i(of)e (the)i(p)r(oten)n(tial)e Fz(V)52 b FA(on)33 b Fz(a)f FA(is)h(nondegenerate)e(as)h(it)i(w)n(as)d(explained)515 822 y(ab)r(o)n(v)n(e.)53 b(In)33 b(the)h(\014rst)f(three)h(examples)e (the)i(p)r(oten)n(tial)f Fz(V)19 b FA(\()p Fz(x)p FA(;)48 b Fz(a)p FA(\))33 b(is)h(real,)g(smo)r(oth)f(in)h Fz(x)515 922 y FA(and)28 b(analytic)h(in)g Fz(a)p FA(.)41 b(The)29 b(function)g Fz(f)9 b FA(\()p Fz(x;)28 b(v)s FA(;)39 b Fz(a)p FA(\))30 b(is)e(real,)h(smo)r(oth)f(in)h Fz(x)h FA(and)f(analytic)f(in)515 1021 y Fz(v)j FA(and)c Fz(a)p FA(.)37 b(Details)28 b(can)f(b)r(e)h(found)g(in)g([Kuk93)n(,)g(Kuk94)n (,)g(Kuk00)o(,)f(Kuk98)o(].)515 1148 y Fp(Example)37 b FA(5.2)p Fp(.)k FA(Nonlinear)26 b(Sc)n(hr\177)-42 b(odinger)26 b(equation)h(\(NLS\),)i(cf.)37 b(Example)27 b(2.8:)1174 1315 y(_)-38 b Fz(u)23 b FA(=)g Fz(i)p FA(\()p Fy(\000)p Fz(u)1492 1327 y Fw(xx)1589 1315 y FA(+)18 b Fz(V)g FA(\()p Fz(x)p FA(;)38 b Fz(a)p FA(\))p Fz(u)18 b FA(+)g Fz("f)9 b FA(\()p Fz(x;)28 b Fy(j)p Fz(u)p Fy(j)2416 1281 y Fv(2)2453 1315 y FA(;)37 b Fz(a)p FA(\))p Fz(u)p FA(\))p Fz(;)516 b FA(\(5.9\))1022 1440 y Fz(u)22 b FA(=)h Fz(u)p FA(\()p Fz(t;)28 b(x)p FA(\))p Fz(;)42 b(x)23 b Fy(2)h FA([0)p Fz(;)j(\031)s FA(];)97 b Fz(u)p FA(\()p Fz(t;)28 b FA(0\))23 b Fy(\021)f Fz(u)p FA(\()p Fz(t;)28 b(\031)s FA(\))c Fy(\021)e FA(0)p Fz(:)461 b FA(\(5.10\))515 1607 y(No)n(w)20 b Fz(d)740 1619 y Fw(J)810 1607 y FA(=)j(0,)f Fz(d)1028 1619 y Fw(A)1105 1607 y FA(=)h(2,)1295 1585 y(~)1280 1607 y Fz(d)g FA(=)g Fz(d)1477 1619 y Fw(H)1563 1607 y FA(=)g(0)d(and)h(w)n(e)g(view)g(the)h(Diric)n(hlet)f(b)r(oundary)g (conditions)515 1706 y(as)31 b(the)h(o)r(dd)g(p)r(erio)r(dic)g(ones)f (\(cf.)51 b(Example)31 b(2.9\).)49 b(The)32 b(theorem)f(applies)g(in)i (the)f(scale)515 1806 y(of)f(o)r(dd)h(p)r(erio)r(dic)f(functions)h (with)g Fz(d)1701 1818 y Fv(0)1767 1806 y FA(=)d(1)i(or)g(2.)48 b(If)32 b Fz(f)40 b FA(is)31 b(ev)n(en)g(and)g(2)p Fz(\031)s FA(-p)r(erio)r(dic)g(in)g Fz(x)p FA(,)515 1906 y(then)k(the)g(theorem)f (applies)h(with)g(an)n(y)f Fz(d)1879 1918 y Fv(0)1951 1906 y Fy(\025)h FA(1)f(and)h(the)g(constructed)f(quasip)r(erio)r(dic) 515 2005 y(solutions)27 b(are)f(smo)r(oth.)p 3318 2005 4 57 v 3322 1953 50 4 v 3322 2005 V 3372 2005 4 57 v 515 2132 a Fp(Example)37 b FA(5.3)p Fp(.)k FA(Nonlinear)26 b(string)h(equation:)37 b Fz(w)r FA(\()p Fz(t;)14 b(x)p FA(\))29 b(satis\014es)e(\(5.10\))g(and)1315 2299 y(\177)-59 b Fz(w)25 b FA(=)e Fz(w)1529 2311 y Fw(xx)1627 2299 y Fy(\000)18 b Fz(V)h FA(\()p Fz(x)p FA(;)38 b Fz(a)p FA(\))p Fz(w)21 b FA(+)d Fz("f)9 b FA(\()p Fz(x;)28 b(w)r FA(;)38 b Fz(a)p FA(\))p Fz(;)515 2466 y FA(where)d(no)n(w)h Fz(V)56 b(>)37 b FA(0)f(and)g Fz(f)9 b FA(\()p Fz(x;)14 b(w)r FA(\))38 b(=)f(0)f(if)g Fz(w)k FA(=)d(0)f(or)f Fz(x)j FA(=)f(0.)62 b(Let)36 b(us)g(denote)g Fz(U)46 b FA(=)515 2566 y(\()p Fz(u;)14 b Fy(\000)p FA(\()p Fy(\000)p FA(\001\))895 2536 y Fu(\000)p Fv(1)p Fw(=)p Fv(2)1065 2566 y FA(_)-37 b Fz(u)o FA(\).)34 b(It)20 b(is)e(a)h(matter)g(of)f (direct)h(v)n(eri\014cation)f(that)h Fz(U)27 b FA(satis\014es)18 b(a)h(semilinear)515 2666 y(Hamiltonian)h(equation)h(\(5.1\))f(in)h(a)g (suitable)f(symplectic)h(Hilb)r(ert)h(scale,)f(formed)f(b)n(y)h(o)r(dd) 515 2765 y(p)r(erio)r(dic)34 b(Sob)r(olev)f(v)n(ector-functions)f (\(cf.)57 b(equation)33 b(\(4.6\)\).)56 b(No)n(w)33 b Fz(d)2807 2777 y Fw(A)2895 2765 y FA(=)g(1,)i Fz(d)3136 2777 y Fw(J)3216 2765 y FA(=)e(0,)529 2843 y(~)515 2865 y Fz(d)23 b FA(=)g Fz(d)712 2877 y Fw(H)798 2865 y FA(=)g Fy(\000)p FA(1.)36 b(Cf.)h([W)-7 b(a)n(y90)n(])28 b(and)g([Bou94)n(,)g (Bam99b)n(].)p 3318 2865 V 3322 2812 50 4 v 3322 2865 V 3372 2865 4 57 v 515 2992 a Fp(Example)37 b FA(5.4)p Fp(.)k FA(KdV-t)n(yp)r(e)27 b(equations:)678 3197 y(_)-38 b Fz(u)23 b FA(=)855 3141 y Fz(@)p 831 3178 97 4 v 831 3254 a(@)5 b(x)937 3197 y FA(\()p Fy(\000)p Fz(u)1082 3209 y Fw(xx)1180 3197 y FA(+)18 b Fz(V)g FA(\()p Fz(x)p FA(;)38 b Fz(a)p FA(\))p Fz(u)18 b FA(+)g Fz("f)9 b FA(\()p Fz(x;)28 b(u)p FA(;)37 b Fz(a)p FA(\)\);)97 b Fz(x)23 b Fy(2)h Fz(S)2454 3163 y Fv(1)2491 3197 y Fz(;)2551 3084 y Fq(Z)2597 3273 y Fw(S)2641 3256 y Ft(1)2691 3197 y Fz(u)14 b(dx)23 b Fy(\021)g FA(0)p Fz(;)148 b FA(\(5.11\))515 3426 y(cf.)37 b(Example)27 b(2.7.)36 b(No)n(w)27 b Fz(d)1380 3438 y Fw(J)1450 3426 y FA(=)22 b(1,)27 b Fz(d)1672 3438 y Fw(A)1750 3426 y FA(=)22 b(2,)1944 3404 y(~)1930 3426 y Fz(d)h FA(=)g Fz(d)2127 3438 y Fw(H)2213 3426 y FA(=)f(0.)p 3318 3426 4 57 v 3322 3374 50 4 v 3322 3426 V 3372 3426 4 57 v 639 3597 a(Theorem)32 b(5.1)h(also)f(applies)g(if)i Fz(x)e Fy(2)h Fx(R)1892 3566 y Fv(1)1968 3597 y FA(and)g(the)g(p)r (oten)n(tial)g Fz(V)19 b FA(\()p Fz(x)p FA(;)47 b Fz(a)p FA(\))33 b(gro)n(ws)e(su\016-)515 3696 y(cien)n(tly)c(fast)h(when)g Fz(x)23 b Fy(!)g(1)p FA(.)515 3823 y Fp(Example)37 b FA(5.5)p Fp(.)k FA(Nonlinear)26 b(Sc)n(hr\177)-42 b(odinger)26 b(equation)h(on)h(the)g(line:)750 3990 y(_)-38 b Fz(u)23 b FA(=)f Fz(i)p FA(\()p Fy(\000)p Fz(u)1067 4002 y Fw(xx)1164 3990 y FA(+)c(\()p Fz(x)1326 3956 y Fv(2)1383 3990 y FA(+)g Fz(\026x)1563 3956 y Fv(4)1619 3990 y FA(+)g Fz(V)h FA(\()p Fz(x)p FA(;)38 b Fz(a)p FA(\)\))p Fz(u)18 b FA(+)g Fz("f)9 b FA(\()p Fy(j)p Fz(u)p Fy(j)2381 3956 y Fv(2)2417 3990 y FA(;)37 b Fz(a)p FA(\))p Fz(u)p FA(\))p Fz(;)97 b(\026)23 b(>)g FA(0)p Fz(;)1358 4115 y(u)g FA(=)f Fz(u)p FA(\()p Fz(t;)28 b(x)p FA(\))p Fz(;)97 b(x)24 b Fy(2)f Fx(R)p Fz(;)103 b(u)23 b Fy(!)g FA(0)82 b(as)h Fy(j)p Fz(x)p Fy(j)23 b(!)g(1)p Fz(:)515 4282 y FA(Here)32 b(the)h(p)r(oten)n (tial)g Fz(V)52 b FA(is)33 b(smo)r(oth,)h(analytic)e(in)h Fz(a)g FA(and)g(v)-5 b(anishes)32 b(as)g Fy(j)p Fz(x)p Fy(j)g(!)g(1)p FA(.)53 b(The)515 4381 y(real-v)-5 b(alued)31 b(function)j Fz(f)42 b FA(is)33 b(analytic.)52 b(No)n(w)32 b Fz(d)2046 4393 y Fw(J)2125 4381 y FA(=)g(0,)i Fz(d)2364 4393 y Fw(A)2450 4381 y FA(=)e(4)p Fz(=)p FA(3,)h Fz(d)2772 4393 y Fw(H)2867 4381 y FA(=)f(0.)52 b(Another)515 4481 y(example)27 b(of)g(this)h(sort)f(see)g(in)h([Kuk93)o(],)g(section)f (2.5.)p 3318 4481 V 3322 4428 50 4 v 3322 4481 V 3372 4481 4 57 v 639 4608 a(The)40 b(time-quasip)r(erio)r(dic)e(solutions,)k (constructed)d(in)g(Examples)f(5.2)h({)g(5.5,)i(are)515 4707 y(linearly)34 b(stable.)60 b(Therefore)34 b(they)i(should)f(b)r(e) h(observ)-5 b(able)34 b(in)i(n)n(umerical)e(mo)r(dels)i(for)515 4807 y(the)d(corresp)r(onding)f(equations.)53 b(Indeed,)35 b(quasip)r(erio)r(dic)e(b)r(eha)n(viour)f(of)h(solutions)g(for)515 4907 y(1D)24 b(HPDEs)h(with)g(small)f(nonlinearit)n(y)f(w)n(as)h (observ)n(ed)f(in)i(man)n(y)f(exp)r(erimen)n(ts,)h(starting)515 5006 y(from)i(the)h(famous)f(n)n(umerics)g(of)h(F)-7 b(ermi{P)n(asta{Ulam)24 b([FPU65];)j(e.g.,)h(see)f([ZIS79)o(].)1905 5255 y(18)p eop PStoPSsaved restore %%Page: (18,19) 10 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 19 18 bop 515 523 a Fs(5.3)112 b(Multiple)35 b(sp)s(ectrum)515 676 y FA(In)24 b(Examples)g(5.2,)f(5.3)h(the)h(equations)e(are)h (considered)f(under)h(the)h(Diric)n(hlet)g(b)r(oundary)515 776 y(conditions.)36 b(If)28 b(w)n(e)f(replace)g(them)h(b)n(y)f(the)h (p)r(erio)r(dic)g(ones)1543 940 y Fz(u)p FA(\()p Fz(t;)g(x)p FA(\))23 b Fy(\021)g Fz(u)p FA(\()p Fz(t;)28 b(x)18 b FA(+)g(2)p Fz(\031)s FA(\))p Fz(;)515 1104 y FA(then)25 b(Theorem)f(5.1)g(w)n(ould)h(not)g(apply)f(since)h(no)n(w)f(the)i (frequencies)e(of)h(the)g(corresp)r(ond-)515 1204 y(ing)f(linear)g (equations)f(are)h(asymptotically)f(double:)35 b(they)25 b(ha)n(v)n(e)e(the)i(form)f Fy(f)p Fz(\025)3030 1168 y Fu(\006)3030 1227 y Fw(j)3086 1204 y Fz(;)14 b(j)28 b Fy(\025)23 b FA(1)p Fy(g)p FA(,)515 1318 y(where)29 b Fy(j)p Fz(\025)828 1283 y Fv(+)828 1341 y Fw(j)904 1318 y Fy(\000)20 b Fz(\025)1037 1283 y Fu(\000)1037 1341 y Fw(j)1093 1318 y Fy(j)27 b(!)h FA(0)h(as)h Fz(j)i Fy(!)27 b(1)p FA(.)45 b(It)30 b(is)g(clear)f(that)h(the)h(n)n(um)n(b)r (ers)e Fy(f)p Fz(\025)2891 1283 y Fu(\006)2891 1341 y Fw(j)2947 1318 y Fy(g)h FA(cannot)g(b)r(e)515 1418 y(re-ordered)25 b(to)i(meet)h(the)f(sp)r(ectral)g(asymptotic)g(condition)g(H2\).)37 b(Still,)28 b(for)f(some)f(semi-)515 1517 y(linear)c(equations)g (\(5.1\))g(assertions)f(of)i(the)g(theorem)g(remain)f(true)h(if)g(the)g (frequencies)g Fz(\025)3344 1529 y Fw(j)515 1617 y FA(are)30 b(not)h(single,)h(but)g(asymptotically)e(they)h(ha)n(v)n(e)f(the)i (same)e(m)n(ultiplicit)n(y)i Fz(m)d Fy(\025)f FA(2)j(and)515 1717 y(b)r(eha)n(v)n(e)e(regularly)-7 b(.)43 b(Corresp)r(onding)29 b(result)h(is)g(pro)n(v)n(ed)e(b)n(y)i(Chierc)n(hia{Y)-7 b(ou)29 b(in)i([CY00)o(],)515 1816 y(using)h(the)i(sc)n(heme,)g (explained)f(in)g(section)g(5.1.)52 b(W)-7 b(e)34 b(do)f(not)g(giv)n(e) f(precise)g(statemen)n(t)515 1916 y(of)i(their)h(theorem,)h(but)g(note) e(that)h(it)h(applies)e(to)h(the)g(nonlinear)f(string)g(equation)g(in) 515 2016 y(Examples)28 b(5.3)g(under)h(the)h(p)r(erio)r(dic)f(b)r (oundary)f(conditions.)42 b(The)29 b(result)g(is)g(the)h(same:)515 2115 y(if)c(the)g(non-degeneracy)d(condition)i(holds,)h(then)g(for)f Fz(")g FA(small)g(enough)g(and)g(for)g(most)h(\(in)515 2215 y(the)h(sense)f(of)h(measure\))f(v)-5 b(alues)26 b(of)h(the)g Fz(n)p FA(-dimensional)f(parameter)f Fz(a)p FA(,)i(solutions)f(of)h(the)515 2314 y(linear)19 b(equation)h(\(5.3\))g (whic)n(h)g(\014ll)h(in)f(a)g(torus)f Fz(T)2021 2284 y Fw(n)2066 2314 y FA(\()p Fz(I)7 b FA(\),)22 b Fz(I)30 b Fy(2)24 b Fx(R)2417 2284 y Fw(n)2417 2335 y Fv(+)2478 2314 y FA(,)d(p)r(ersist)f(as)g(linearly)f(stable)515 2414 y(time-quasip)r(erio)r(dic)27 b(solutions)f(of)i(the)g(corresp)r (onding)e(non-linear)g(equation)h(\(5.1\).)639 2514 y(W)-7 b(e)25 b(note)e(that)i(earlier)d(this)i(p)r(ersistence)g(result)f(w)n (as)g(pro)n(v)n(en)g(b)n(y)g(Bourgain)f([Bou94)o(],)515 2613 y(who)27 b(used)h(another)e(KAM-sc)n(heme,)h(discussed)g(in)h(the) g(next)g(section.)515 2843 y Fs(5.4)112 b(Space-m)m(ultidimensional)35 b(problems)515 2996 y FA(The)d(abstract)f(Theorem)g(5.1)h(is)g(a)f (\015exible)h(to)r(ol)g(to)g(study)h(1D)f(HPDEs,)g(but)h(it)g Fp(never)515 3095 y FA(applies)28 b(to)h(space-m)n(ultidimensional)e (equations)h(since)h(the)g(sp)r(ectral)g(assumption)f(H2\))515 3195 y(nev)n(er)e(holds)h(in)h(high)f(dimensions.)37 b(A)n(t)27 b(the)h(time)g(of)f(this)h(writing,)f(the)h(only)f (published)515 3295 y(KAM)33 b(result,)h(whic)n(h)f(applies)g(to)f (higher-dimensional)g(HPDEs,)i(is)f(due)g(to)g(Bourgain)515 3394 y([Bou98)n(].)j(In)22 b(that)h(w)n(ork)f(the)h(2D)f(NLS)h (equation)f(as)g(in)h(the)g(Example)f(2.8)g(is)g(considered.)515 3494 y(F)-7 b(or)22 b(tec)n(hnical)g(reasons)f(the)i(p)r(oten)n(tial)g (term)g Fz(V)c(u)j FA(is)h(replaced)e(there)i(b)n(y)f(the)i(con)n(v)n (olution)515 3594 y Fz(V)37 b Fy(\003)18 b Fz(u)p FA(:)684 3780 y(_)-38 b Fz(u)23 b FA(=)f Fz(i)856 3713 y Fq(\000)912 3780 y Fy(\000)c FA(\001)p Fz(u)h FA(+)f Fz(V)g FA(\()p Fz(x)p FA(;)38 b Fz(a)p FA(\))19 b Fy(\003)f Fz(u)g FA(+)g Fz(")1796 3724 y(@)p 1773 3761 97 4 v 1773 3837 a(@)10 b FA(\026)-47 b Fz(u)1879 3780 y(g)s FA(\()p Fz(u;)32 b FA(\026)-47 b Fz(u)o FA(\))2131 3713 y Fq(\001)2170 3780 y Fz(;)96 b(u)23 b FA(=)g Fz(u)p FA(\()p Fz(t;)k(x)p FA(\))p Fz(;)38 b(x)23 b Fy(2)h Fx(T)2953 3746 y Fv(2)2989 3780 y Fz(:)155 b FA(\(5.12\))515 3972 y(The)32 b(p)r(oten)n(tial)f Fz(V)19 b FA(\()p Fz(x)p FA(;)45 b Fz(a)p FA(\))32 b(is)g(real)f (analytic)g(and)g Fz(g)s FA(\()p Fz(u;)i FA(\026)-48 b Fz(u)p FA(\))32 b(is)g(a)f(real-v)-5 b(alued)31 b(p)r(olynomial)515 4072 y(of)h Fz(u)f FA(and)38 b(\026)-48 b Fz(u)p FA(.)50 b(This)32 b(equation)g(has)f(the)i(form)f(\(5.1\))o(,)i(where)d Fz(Au)f FA(=)h Fy(\000)p FA(\001)p Fz(u)20 b FA(+)h Fz(V)41 b Fy(\003)21 b Fz(u)31 b FA(and)515 4172 y Fz(J)8 b(u)27 b FA(=)h Fz(iu)p FA(.)46 b(The)30 b(basis)g Fy(f)p Fz(')1359 4184 y Fw(k)1400 4172 y Fy(g)g FA(as)g(in)h(\(5.2\))f(is)h(formed)f(b)n (y)h(normalised)e(exp)r(onen)n(ts)h Fy(f)p Fz(e)3263 4142 y Fw(is)p Fu(\001)p Fw(x)515 4271 y FA(and)d Fz(ie)744 4241 y Fw(is)p Fu(\001)p Fw(x)874 4271 y Fz(;)14 b(s)22 b Fy(2)i Fx(Z)1112 4241 y Fv(2)1143 4271 y Fy(g)p FA(,)k(re-n)n (umerated)e(prop)r(erly)-7 b(,)27 b(and)1396 4436 y Fz(\025)1444 4401 y Fw(J)1444 4456 y(s)1514 4436 y Fy(\021)c FA(1)p Fz(;)96 b(\025)1811 4401 y Fw(A)1811 4456 y(s)1889 4436 y FA(=)22 b Fy(j)p Fz(s)p Fy(j)2061 4401 y Fv(2)2117 4436 y FA(+)2213 4415 y(^)2200 4436 y Fz(V)d FA(\()p Fz(s)p FA(;)37 b Fz(a)p FA(\))p Fz(;)515 4600 y FA(where)22 b Fy(f)805 4579 y FA(^)792 4600 y Fz(V)c FA(\()p Fz(s)p FA(;)37 b Fz(a)p FA(\))p Fy(g)22 b FA(are)g(the)h(F)-7 b(ourier)21 b(co)r(e\016cien)n(ts)h(of)g Fz(V)d FA(.)35 b(F)-7 b(or)22 b(an)n(y)g Fz(n)p FA(,)h(the)g(linear)f(equation)515 4699 y(\(5.12\))o Fy(j)750 4711 y Fw(")p Fv(=0)897 4699 y FA(has)27 b(quasip)r(erio)r(dic)g(solutions)1535 4930 y Fz(u)c FA(=)1733 4826 y Fw(n)1693 4851 y Fq(X)1696 5028 y Fw(j)s Fv(=1)1827 4930 y Fz(z)1866 4942 y Fw(s)1897 4950 y Fn(j)1932 4930 y Fz(e)1971 4883 y Fw(i\025)2033 4858 y Fn(A)2033 4900 y(s)2061 4913 y(j)2097 4883 y Fw(t)2126 4930 y Fz(')2180 4942 y Fw(s)2211 4950 y Fn(j)2247 4930 y FA(\()p Fz(x)p FA(\))809 b(\(5.13\))1905 5255 y(19)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 20 19 bop 515 523 a FA(\(these)35 b(are)f(tra)5 b(jectories)33 b(of)h(the)h(equation)g(\(5.6\))f(on)h(the)g Fz(n)p FA(-torus)e (\(5.5\),)k(where)d Fz(I)3244 535 y Fw(j)3314 523 y FA(=)525 590 y Fv(1)p 525 604 34 4 v 525 651 a(2)568 623 y Fy(j)p Fz(z)630 635 y Fw(s)661 643 y Fn(j)696 623 y Fy(j)719 593 y Fv(2)790 623 y FA(and)g Fz(I)994 635 y Fw(s)1063 623 y FA(=)f(0)h(if)g Fz(s)g FA(di\013ers)g(from)f(all)h Fz(s)2009 635 y Fw(j)2044 623 y FA(\).)56 b(F)-7 b(or)33 b(simplicit)n(y)h(let)g(us)g(assume)f(that)515 738 y Fz(a)559 750 y Fw(j)617 738 y FA(=)717 717 y(^)704 738 y Fz(V)19 b FA(\()p Fz(s)842 750 y Fw(j)877 738 y FA(;)37 b Fz(a)p FA(\))p Fz(;)97 b(j)28 b FA(=)23 b(1)p Fz(;)14 b(:)g(:)g(:)f(;)h(n)p FA(.)35 b(Then)24 b(the)f(result)g(of)g([Bou98)n (])h(is)f(that)g(for)g(most)f(v)-5 b(alues)515 838 y(of)31 b(the)h(parameter)e Fz(a)i FA(\(in)g(the)g(same)f(sense)g(as)g(in)h (Theorem)f(5.1\),)h(the)g(solution)f(\(5.13\))515 938 y(p)r(ersists)j(as)h(a)f(time-quasip)r(erio)r(dic)g(solution)h(of)g (the)g(equation)g(\(5.12\))o(.)59 b(In)35 b(di\013erence)515 1037 y(with)28 b(the)g(1D)f(case,)g(it)h(is)g(unkno)n(wn)f(if)h(the)g (p)r(ersisted)f(solutions)g(are)g(linearly)g(stable.)639 1137 y(Presumably)-7 b(,)24 b(in)h(the)g(nearest)e(future)i(similar)f (results)g(will)g(b)r(e)h(obtained)f(for)g(a)g(n)n(um-)515 1237 y(b)r(er)34 b(of)g(other)g(semilinear)f(equations.)56 b(W)-7 b(e)34 b(do)g(not)g(exp)r(ect)h(that)f(an)g(abstract)f(KAM-)515 1336 y(theorem,)i(whic)n(h)e(applies)h(to)f(di\013eren)n(t)h(classes)e (of)i(space-m)n(ultidimensional)e(HPDEs,)515 1436 y(will)f(b)r(e)h(pro) n(v)n(en.)46 b(Certainly)31 b(it)h(will)f(b)r(e)h(m)n(uc)n(h)f(more)f (di\016cult)j(to)e(handle)g(quasi-linear)515 1535 y(equations)26 b(that)i(are)f(not)h(semi-linear.)639 1685 y(The)36 b(pro)r(of)f(in)h ([Bou98)o(])g(is)g(based)f(on)g(a)h(KAM-sc)n(heme,)h(di\013eren)n(t)f (from)f(that)h(de-)515 1785 y(scrib)r(ed)21 b(in)h(section)g(5.1.)34 b(Originally)20 b(this)i(sc)n(heme)f(is)h(due)g(to)f(Craig)g(and)g(W)-7 b(a)n(yne)22 b([CW93)o(])515 1884 y(who)32 b(used)g(it)g(to)g (construct)g(p)r(erio)r(dic)g(solutions)f(of)h(nonlinear)f(w)n(a)n(v)n (e)g(equations.)49 b(Also)515 1984 y(see)27 b([Bou94)n(].)639 2083 y(No)n(w)34 b(w)n(e)g(brie\015y)f(describ)r(e)h(the)h(sc)n(heme,)g (using)f(the)g(notations)f(from)h(section)g(5.1.)515 2183 y(When)39 b(the)f(p)r(erturbation)g Fz("H)1527 2195 y Fv(1)1603 2183 y FA(is)g(decomp)r(osed)g(as)f(in)i(\(5.8\),)i(w)n(e)d (extract)f(the)i(term)515 2283 y Fz(")p Fy(h)p Fz(h)634 2253 y Fw(y)r(y)709 2283 y FA(\()p Fz(q)s FA(\))p Fz(y)s(;)28 b(y)s Fy(i)36 b FA(from)f Fz("H)1339 2253 y Fv(1)1332 2303 y(1)1411 2283 y FA(and)g(add)g(it)h(to)f(the)g(in)n(tegrable)f (part)h Fz(H)2749 2295 y Fv(0)2786 2283 y FA(.)60 b(After)36 b(this)g(the)515 2382 y(hamiltonian)25 b(to)g(b)r(e)h(killed)g(is)f (the)h(sum)g(of)f(the)h(three)g(terms)f Fz(h)p FA(\()p Fz(q)s FA(\))15 b(+)f Fz(h)2769 2352 y Fw(p)2807 2382 y FA(\()p Fz(q)s FA(\))h(+)f Fy(h)p Fz(h)3085 2352 y Fw(y)3125 2382 y FA(\()p Fz(q)s FA(\))p Fz(;)28 b(y)s Fy(i)p FA(;)515 2482 y(accordingly)35 b(the)i(hamiltonian)g Fz(F)49 b FA(is)36 b(a)h(sum)g(of)g(three)g(terms)f(as)h(w)n(ell.)64 b(W)-7 b(e)37 b(ha)n(v)n(e)f(to)515 2582 y(\014nd)i(them)h(from)e(the)i (\014rst)e(three)h(homological)e(equations.)67 b(The)38 b(\014rst)g(t)n(w)n(o)f(are)g(not)515 2681 y(di\016cult,)g(but)f(the)f (third)h(one)e(is)h(a)g(real)f(problem)g(since)h(the)g(op)r(erator)f Fz(A)h FA(is)g(not)g(an)n(y)515 2781 y(more)25 b(constan)n(t-co)r (e\016cien)n(t)g(but)i(equals)f Fz(A)1906 2793 y Fv(0)1959 2781 y FA(+)2062 2760 y(^)2040 2781 y Fz(A)p FA(\()p Fz(q)s FA(\),)h(where)2517 2760 y(^)2495 2781 y Fz(A)g FA(is)f(a)g(b)r(ounded)h(op)r(erator)515 2880 y(of)k(order)f Fz(")h FA(\(it)h(c)n(hanges)e(from)h(one)g(KAM-step)g(to)g(another\).) 47 b(The)31 b(resolution)g(of)g(this)515 2980 y(equation)h(for)f(high)i (KAM)f(steps)g(is)h(the)f(most)h(di\016cult)g(part)f(of)g(implemen)n (tation)h(the)515 3080 y(Craig{W)-7 b(a)n(yne{Bourgain)23 b(KAM-sc)n(heme.)515 3312 y Fs(5.5)112 b(P)m(erturbations)36 b(of)i(in)m(tegrable)e(equations)515 3465 y FA(Let)c(us)g(consider)e(a) i(quasilinear)e(HPDE)i(on)g(a)f(\014nite)i(space-in)n(terv)-5 b(al,)31 b(whic)n(h)h(is)g(an)f(in-)515 3565 y(tegrable)36 b(Hamiltonian)i(equation)f(\(4.1\))g(in)h(some)f(symplectic)g(Hilb)r (ert)i(scale)d(\()p Fy(f)p Fz(X)3279 3577 y Fw(s)3314 3565 y Fy(g)p FA(,)515 3665 y Fz(\013)568 3677 y Fv(2)628 3665 y FA(=)736 3644 y(\026)716 3665 y Fz(J)8 b(dx)r Fy(^)r Fz(dx)p FA(\).)37 b(As)20 b(w)n(e)f(explained)g(in)h(section)f (4.1,)h(this)g(equation)f(has)g(in)n(v)-5 b(arian)n(t)19 b(\014nite-)515 3764 y(gap)32 b(symplectic)g(manifolds)h Fy(T)1538 3734 y Fv(2)p Fw(n)1649 3764 y FA(suc)n(h)f(that)h (restriction)f(of)39 b(\(4.1\))32 b(to)h(an)n(y)e(of)i(them)g(is)515 3864 y(in)n(tegrable.)i(In)26 b(this)h(section)e(w)n(e)h(discuss)g(the) h(results)e(on)h(p)r(ersistence)g(of)g(quasip)r(erio)r(dic)515 3963 y(solutions)32 b(that)h(\014ll)g(in)g(these)g(manifolds,)h(pro)n (vided)e(b)n(y)h(the)g(KAM)g(for)g(PDEs)f(theory)-7 b(.)515 4063 y(W)g(e)28 b(shall)g(see)g(that)g(they)g(are)g(v)n(ery)f(similar)g (to)h(the)g(celebrated)g(Kolmogoro)n(v)d(theorem,)515 4163 y(whic)n(h)35 b(states)g(that)g Fp(most)i(of)g(quasip)l(erio)l (dic)j(solutions)d(of)g(a)h(nonde)l(gener)l(ate)f(analytic)515 4262 y(inte)l(gr)l(able)d(\(\014nite-dimensional\))h(Hamiltonian)g (system)e(p)l(ersist)h(under)g(smal)t(l)h(p)l(ertur-)515 4362 y(b)l(ations)d(of)g(the)g(hamiltonian)6 b FA(;)33 b(see)c([Arn63)o(,)h(MS71)o(])g(and)f(Addendum)i(in)f([Kuk00)o(].)43 b(W)-7 b(e)515 4462 y(state)27 b(the)h(main)g(result)f(as)g(a)515 4628 y FB(Theorem)21 b(5.6)i(\(Metatheorem\).)32 b Fp(Most)24 b(of)f(quasip)l(erio)l(dic)j(solutions)d(that)g(\014l)t(l)g(in)g(any) 515 4727 y(\014nite-gap)36 b(manifold)j Fy(T)1305 4697 y Fv(2)p Fw(n)1420 4727 y Fp(as)e(ab)l(ove)g(p)l(ersist)g(under)f(smal) t(l)h(Hamiltonian)h(quasiline)l(ar)515 4827 y(analytic)e(p)l(erturb)l (ations)f(of)h(the)f(inte)l(gr)l(able)h(e)l(quation.)55 b(If)36 b(the)f(\014nite-gap)h(solutions)f(in)515 4926 y Fy(T)581 4896 y Fv(2)p Fw(n)689 4926 y Fp(ar)l(e)30 b(line)l(arly)h(stable,)g(then)e(the)h(p)l(ersiste)l(d)h(solutions)f (ar)l(e)g(line)l(arly)h(stable)f(as)g(wel)t(l.)1905 5255 y FA(20)p eop PStoPSsaved restore %%Page: (20,21) 11 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 21 20 bop 639 523 a FA(In)23 b(the)f(assertion)f(ab)r(o)n(v)n(e)g(the)i (statemen)n(t)f(`most)g(of)g(quasip)r(erio)r(dic)g(solutions)f(p)r (ersist')515 623 y(means)29 b(the)h(follo)n(wing.)43 b(Due)30 b(to)f(the)i(Liouville{Arnold)d(theorem)h([Arn89)o(,)h(HZ94)o (],)h(the)515 722 y(manifold)26 b Fy(T)923 692 y Fv(2)p Fw(n)1027 722 y FA(can)f(b)r(e)i(co)n(v)n(ered)c(b)n(y)j(c)n(harts,)f (di\013eomorphic)g(to)h Fz(B)19 b Fy(\002)14 b Fx(T)2801 692 y Fw(n)2869 722 y FA(=)22 b Fy(f)p Fz(p;)28 b(q)s Fy(g)d FA(\()p Fz(B)30 b FA(is)515 822 y(a)f(ball)g(in)h Fx(R)902 792 y Fw(n)953 822 y FA(\),)g(with)g(c)n(hart-maps)e(\010)1723 834 y Fv(0)1786 822 y FA(:)f Fz(B)c Fy(\002)d Fx(T)2063 792 y Fw(n)2133 822 y Fy(!)27 b(T)2309 792 y Fv(2)p Fw(n)2417 822 y FA(suc)n(h)i(that)h(\010)2848 792 y Fu(\003)2848 843 y Fv(0)2886 822 y Fz(\013)2939 834 y Fv(2)3002 822 y FA(=)c Fz(dp)20 b Fy(^)g Fz(dq)s FA(,)515 922 y(and)j(the)h(curv)n (es)f(\010)1122 934 y Fv(0)1159 922 y FA(\()p Fz(p;)28 b(q)14 b FA(+)d Fz(t)p Fy(r)p Fz(h)p FA(\()p Fz(p)p FA(\)\))23 b(are)g(solutions)g(of)g(the)h(in)n(tegrable)f(equation,)h(where)515 1021 y Fz(h)p FA(\()p Fz(p)p FA(\))34 b(=)h Fz(H)29 b Fy(\016)23 b FA(\010)1026 1033 y Fv(0)1063 1021 y FA(\()p Fz(p;)28 b(q)s FA(\).)58 b(Let)35 b(us)f(denote)h(b)n(y)f Fz(")g FA(the)h(small)f(co)r(e\016cien)n(t)g(in)h(fron)n(t)f(of)h(the) 515 1121 y(p)r(erturbation.)49 b(Then)32 b(for)f(ev)n(ery)g(c)n(hart)g (there)h(exists)f(a)h(Borel)e(subset)i Fz(B)2912 1133 y Fw(")2978 1121 y Fy(\032)e Fz(B)36 b FA(and)31 b(a)515 1220 y(map)c(\010)759 1232 y Fw(")818 1220 y FA(:)c Fz(B)927 1232 y Fw(")981 1220 y Fy(\002)18 b Fx(T)1120 1190 y Fw(n)1187 1220 y Fy(!)24 b Fz(X)1363 1232 y Fw(d)1429 1220 y FA(\()p Fz(d)k FA(is)f(\014xed\),)h(with)g(the)g(follo)n(wing)f (prop)r(erties:)639 1320 y(i\))h(mes\()p Fz(B)t Fy(n)p Fz(B)1065 1332 y Fw(")1101 1320 y FA(\))23 b Fy(!)g FA(0)k(as)g Fz(")c Fy(!)g FA(0;)639 1420 y(ii\))33 b(the)f(map)g(\010)1146 1432 y Fw(")1212 1420 y FA(:)e Fz(B)1328 1432 y Fw(")1385 1420 y Fy(\002)21 b Fx(T)1527 1390 y Fw(n)1602 1420 y Fy(!)31 b Fz(X)1785 1432 y Fw(d)1855 1420 y FA(is)h Fz(C)2008 1360 y Fy(p)p 2077 1360 39 4 v 60 x Fz(")p FA(-close)f(to)h(\010)2513 1432 y Fv(0)2582 1420 y FA(in)g(the)h(Lipsc)n(hitz)f(norm)515 1519 y(and)27 b(is)h(analytic)f(in)g Fz(q)g Fy(2)c Fx(T)1370 1489 y Fw(n)1415 1519 y FA(;)639 1619 y(iii\))32 b(there)g(exists)f(a)g (map)h Fz(!)1536 1631 y Fw(")1601 1619 y FA(:)e Fz(B)1717 1631 y Fw(")1782 1619 y Fy(!)g Fx(R)1949 1589 y Fw(n)2000 1619 y FA(,)j Fz(C)6 b(")p FA(-close)30 b(to)i(the)g(gradien)n(t)f(map) g Fy(r)p Fz(h)h FA(in)515 1719 y(the)26 b(Lipsc)n(hitz)g(norm,)f(suc)n (h)h(that)g(the)g(curv)n(es)e Fz(t)f Fy(7!)g FA(\010)2223 1731 y Fw(")2259 1719 y FA(\()p Fz(p;)28 b(q)17 b FA(+)e Fz(t!)2600 1731 y Fw(")2635 1719 y FA(\()p Fz(p)p FA(\)\),)27 b Fz(p)c Fy(2)g Fz(B)3029 1731 y Fw(")3065 1719 y FA(,)j Fz(q)g Fy(2)d Fx(T)3311 1688 y Fw(n)3356 1719 y FA(,)515 1818 y(are)j(solutions)h(for)g(the)h(p)r(erturb)r(ed)g(equation.)639 1918 y(The)23 b(statemen)n(t)f(of)h(Theorem)e(5.6)h(is)g(pro)n(v)n(en)f (under)i(a)f(n)n(um)n(b)r(er)g(of)g(assumptions)g(\(see)515 2017 y([Kuk00)n(],)30 b([EKMY02)n(]\).)41 b(These)29 b(assumptions)f(are)g(c)n(hec)n(k)n(ed)f(for)i(suc)n(h)f(basic)g(in)n (tegrable)515 2117 y(HPDEs)h(as)g(KdV,)h(Sine{)g(and)g(Sinh{Gordon)f (equations.)43 b(There)29 b(are)g(no)h(doubts)g(that)515 2217 y(they)18 b(also)g(hold)g(for)g(the)h(Zakharo)n(v)c({)j(Shabat)g (equations)2329 2187 y Fv(7)2385 2217 y FA(\(but)h(the)g(theorem)f(in)g ([Kuk00)o(,)515 2316 y(EKMY02)n(])30 b(do)r(es)f(not)h(apply)f(to)h (the)g(Kadom)n(tsev{P)n(etviash)n(vili)c(equation\).)43 b(Belo)n(w)28 b(w)n(e)515 2416 y(presen)n(t)38 b(a)h(sc)n(heme)g(of)g (the)g(pro)r(of)g(and)g(discuss)g(the)g(restrictions)f(on)h(the)g(in)n (tegrable)515 2516 y(HPDE)27 b(whic)n(h)h(allo)n(w)e(to)i(implemen)n(t) g(it.)639 2615 y(W)-7 b(e)25 b(view)h(\(4.1\))e(as)g(an)g(equation)g (in)g(the)h(Hilb)r(ert)g(space)f Fz(X)2515 2627 y Fw(d)2553 2615 y FA(,)i(and)e(denote)g(the)h(quasi-)515 2715 y(linear)i (hamiltonian)g(of)g(the)h(p)r(erturb)r(ed)g(equation)f(as)1347 2882 y Fz(H)1416 2894 y Fw(")1475 2882 y FA(=)1572 2849 y Fv(1)p 1572 2863 34 4 v 1572 2911 a(2)1615 2882 y Fy(h)p Fz(Ax;)i(x)p Fy(i)19 b FA(+)f Fz(h)2037 2894 y Fv(0)2074 2882 y FA(\()p Fz(x)p FA(\))i(+)e Fz("h)2375 2894 y Fv(1)2412 2882 y FA(\()p Fz(x)p FA(\))p Fz(:)515 3049 y FA(Accordingly)-7 b(,)41 b Fz(H)1076 3061 y Fv(0)1155 3049 y FA(=)1272 3017 y Fv(1)p 1272 3031 V 1272 3078 a(2)1315 3049 y Fy(h)p Fz(Ax;)29 b(x)p Fy(i)e FA(+)e Fz(h)1752 3061 y Fv(0)1829 3049 y FA(is)39 b(the)g(hamiltonian)g Fz(H)46 b FA(of)39 b(the)h(unp)r(erturb)r(ed)515 3149 y(equation)27 b(\(4.1\))o(.)639 3249 y(Step)g(1.)36 b(Let)26 b(us)g(consider)f(an)n(y)g(\014nite-gap)h (solution)f Fz(u)2385 3261 y Fv(0)2422 3249 y FA(\()p Fz(t)p FA(\))f(=)e(\010)2687 3261 y Fv(0)2724 3249 y FA(\()p Fz(p)2798 3261 y Fv(0)2836 3249 y Fz(;)28 b(q)2924 3261 y Fv(0)2976 3249 y FA(+)15 b Fz(t)p Fy(r)p Fz(h)p FA(\()p Fz(p)3277 3261 y Fv(0)3314 3249 y FA(\)\))515 3348 y(and)27 b(linearise)g(\(4.1\))g(ab)r(out)h(it:)1576 3516 y(_)-35 b Fz(v)26 b FA(=)d Fz(J)8 b FA(\()p Fy(r)p Fz(H)f FA(\()p Fz(u)2029 3528 y Fv(0)2066 3516 y FA(\()p Fz(t)p FA(\)\)\))2224 3528 y Fu(\003)2263 3516 y Fz(v)s(:)838 b FA(\(5.14\))515 3683 y(The)22 b(theory)f(of)h(in)n(tegrable)f (equations)g(pro)n(vides)f(to)r(ols)i(to)g(reduce)f(this)i(equation)e (to)h(con-)515 3782 y(stan)n(t)c(co)r(e\016cien)n(ts)g(b)n(y)h(means)f (of)g(a)h(time-quasip)r(erio)r(dic)f(substitution)h Fz(v)s FA(\()p Fz(t)p FA(\))24 b(=)e Fz(G)p FA(\()p Fz(p)3151 3794 y Fv(0)3189 3782 y Fz(;)28 b(q)3277 3794 y Fv(0)3314 3782 y FA(+)515 3882 y Fz(t)p Fy(r)p Fz(h)p FA(\()p Fz(p)736 3894 y Fv(0)773 3882 y FA(\)\))s(~)-45 b Fz(v)t FA(\()p Fz(t)p FA(\),)50 b(where)44 b Fz(G)p FA(\()p Fz(p;)28 b(q)s FA(\),)50 b(\()p Fz(p;)28 b(q)s FA(\))52 b Fy(2)g Fz(B)34 b Fy(\002)c Fx(T)2244 3852 y Fw(n)2288 3882 y FA(,)50 b(is)45 b(a)f(symplectic)h(linear)f(map)515 3982 y Fz(G)p FA(\()p Fz(p;)28 b(q)s FA(\))38 b(:)f Fz(Y)923 3994 y Fw(d)1000 3982 y Fy(!)h Fz(Z)1178 3994 y Fw(d)1252 3982 y FA(\(see)f([Kuk00)n(],)i(sections)d(5,)i(6\).)63 b(Here)36 b Fz(Y)2608 3994 y Fw(d)2683 3982 y FA(is)g(a)g(\014xed)h (symplec-)515 4081 y(tic)27 b(subspace)g(of)g Fz(Z)1132 4093 y Fw(d)1197 4081 y FA(of)g(co)r(dimension)g(2)p Fz(n)p FA(.)36 b(The)28 b(restriction,)e(whic)n(h)h(w)n(e)g(imp)r(ose)g (at)g(this)515 4181 y(step,)k(is)f(that)h(the)f(op)r(erator)f Fz(G)p FA(\()p Fz(p;)f(q)s FA(\))j(is)f(a)g(compact)f(p)r(erturbation)h (of)g(the)h(em)n(b)r(edding)515 4281 y Fz(Y)563 4293 y Fw(d)625 4281 y Fy(!)23 b Fz(Z)788 4293 y Fw(d)826 4281 y FA(,)28 b(whic)n(h)f(analytically)g(dep)r(ends)h(on)f(\()p Fz(p;)h(q)s FA(\).)639 4380 y(Step)g(2.)37 b(The)28 b(map)f Fz(G)h FA(from)f(the)h(Step)g(1)f(de\014nes)h(an)f(analytic)g(map)1588 4548 y Fz(B)c Fy(\002)18 b Fx(T)1813 4513 y Fw(n)1876 4548 y Fy(\002)g Fz(Y)2007 4560 y Fw(d)2069 4548 y Fy(!)23 b Fz(X)2244 4560 y Fw(d)2282 4548 y Fz(;)515 4715 y FA(linear)k(and)g (symplectic)h(in)g Fz(y)d Fy(2)f Fz(Y)1606 4727 y Fw(d)1645 4715 y FA(.)37 b(This)27 b(map)h(de\014nes)f(a)g(symplectomorphism)1066 4882 y Fz(B)c Fy(\002)18 b Fx(T)1291 4848 y Fw(n)1354 4882 y Fy(\002)g Fz(B)1500 4894 y Fw(\016)1536 4882 y FA(\()p Fz(Y)1616 4894 y Fw(d)1655 4882 y FA(\))23 b Fy(!)h Fz(X)1886 4894 y Fw(d)1924 4882 y Fz(;)97 b(B)2107 4894 y Fw(\016)2143 4882 y FA(\()p Fz(Y)2223 4894 y Fw(d)2263 4882 y FA(\))23 b(=)g Fy(fk)p Fz(y)s Fy(k)2576 4894 y Fw(d)2636 4882 y Fz(<)f(\016)s Fy(g)p Fz(;)339 b FA(\(5.15\))p 515 4929 1146 4 v 607 4983 a Fm(7)642 5006 y Fl(See)24 b([GK03)q(])f(for)g(an)h Ff(ad)i(ho)l(c)i Fl(KAM-theorem)22 b(for)h(the)i(defo)r(cusing)f(equation.)1905 5255 y FA(21)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 22 21 bop 515 523 a FA(suc)n(h)31 b(that)h(linearisation)f(in)h Fz(y)j FA(at)c Fz(y)i FA(=)d(0)h(of)h(the)g(latter)g(equals)f(the)h (former)f(\([Kuk00)o(],)515 623 y(section)c(7\).)639 722 y(Step)c(3.)34 b(W)-7 b(e)23 b(use)e(the)i(map)f(\(5.15\))f(to)g (pass)h(in)g(the)g(hamiltonian)g Fz(H)2772 734 y Fw(")2829 722 y FA(to)g(the)g(v)-5 b(ariables)515 822 y(\()p Fz(p;)14 b(q)s(;)g(y)s FA(\).)37 b(Retaining)27 b(linear)g(and)g(quadratic)g(in) h Fz(y)i FA(terms)d(w)n(e)g(get)736 997 y Fz(H)805 1009 y Fw(")841 997 y FA(\()p Fz(p;)h(q)s(;)g(y)s FA(\))23 b(=)f Fz(h)p FA(\()p Fz(p)p FA(\))d(+)1509 964 y Fv(1)p 1509 978 34 4 v 1509 1026 a(2)1552 997 y Fy(hA)p FA(\()p Fz(p)p FA(\))p Fz(y)s(;)28 b(y)s Fy(i)19 b FA(+)f Fz(h)2077 1009 y Fv(3)2114 997 y FA(\()p Fz(p;)28 b(q)s(;)f(y)s FA(\))19 b(+)f Fz("h)2594 1009 y Fv(1)2631 997 y FA(\()p Fz(p;)27 b(q)s(;)h(y)s FA(\))p Fz(;)222 b FA(\(5.16\))515 1172 y(where)36 b Fz(h)812 1184 y Fv(3)887 1172 y FA(=)h Fz(O)r FA(\()p Fy(k)p Fz(y)s Fy(k)1214 1142 y Fv(3)1214 1196 y Fw(d)1252 1172 y FA(\).)64 b(Calculations)36 b(sho)n(w)f(that)i Fz(h)2307 1184 y Fv(3)2344 1172 y FA(\()p Fz(p;)14 b(q)s(;)g(y)s FA(\))37 b(con)n(tains)e(terms)h(suc)n(h)515 1272 y(that)i(their)g (gradien)n(t)e(maps)i(ha)n(v)n(e)f(the)h(same)f(order)g(as)g(the)h(op)r (erator)e Fy(A)p FA(\()p Fz(p)p FA(\).)68 b(If)39 b(this)515 1371 y(really)32 b(w)n(as)h(the)h(case,)h(then)f(the)g(Hamiltonian)g (equation)f(w)n(ould)h(not)f(b)r(e)h(quasilinear,)515 1471 y(whic)n(h)28 b(w)n(ould)g(complicate)g(its)h(study)g(a)f(lot.)39 b(F)-7 b(ortunately)g(,)29 b(this)f(do)r(es)h(not)f(happ)r(en)h(due)515 1571 y(to)g(a)g(cancellation)f(of)i(a)f(v)n(ery)f(general)g(nature)h (\(see)g(Lemma)g(7.5)g(in)g([Kuk00)o(]\),)h(and)f(w)n(e)515 1670 y(ha)n(v)n(e)1511 1770 y(ord)13 b Fy(r)p Fz(h)1762 1782 y Fv(3)1822 1770 y Fz(<)23 b FA(ord)13 b Fy(A)p FA(\()p Fz(p)p FA(\))19 b Fy(\000)f FA(1)p Fz(:)784 b FA(\(5.17\))639 1915 y(Step)43 b(4.)81 b(In)n(v)-5 b(arian)n(t)41 b(tori)h(of)h(the)g(unp)r(erturb)r(ed)f(system)h(with)g(the)f (hamiltonian)515 2014 y Fz(H)584 2026 y Fv(0)621 2014 y FA(\()p Fz(p;)28 b(q)s(;)g(y)s FA(\))g(ha)n(v)n(e)g(the)h(form)f Fy(f)p Fz(p)c FA(=)g(const)p Fz(;)14 b(y)27 b FA(=)e(0)p Fy(g)p FA(.)39 b(Let)29 b(us)f(scale)g(the)h(v)-5 b(ariables)27 b(near)h Fz(a)515 2114 y FA(torus)h Fy(f)p Fz(p)d FA(=)g Fz(a;)14 b(y)29 b FA(=)e(0)p Fy(g)p FA(:)70 b Fz(p)26 b FA(=)h Fz(a)20 b FA(+)f Fz(")1698 2084 y Fv(2)p Fw(=)p Fv(3)1809 2114 y FA(~)-49 b Fz(p)p FA(,)30 b Fz(q)g FA(=)j(~)-49 b Fz(q)t FA(,)30 b Fz(y)f FA(=)e Fz(")2350 2084 y Fv(1)p Fw(=)p Fv(3)2459 2114 y FA(~)-47 b Fz(y)r FA(.)44 b(In)30 b(the)g(scaled)f(v)-5 b(ariables)515 2214 y(the)28 b(p)r(erturb)r(ed)g (equation)f(has)g(the)h(hamiltonian)800 2389 y(const)13 b(+)p Fz(!)s FA(\()p Fz(a)p FA(\))18 b Fy(\001)26 b FA(~)-49 b Fz(p)18 b FA(+)1444 2356 y Fv(1)p 1444 2370 V 1444 2417 a(2)1487 2389 y Fy(hA)p FA(\()p Fz(a)p FA(\))6 b(~)-48 b Fz(y)t(;)33 b FA(~)-47 b Fz(y)r Fy(i)19 b FA(+)f Fz(O)r FA(\()p Fz(")2102 2354 y Fv(1)p Fw(=)p Fv(3)2207 2389 y FA(\))p Fz(;)97 b(!)s FA(\()p Fz(a)p FA(\))23 b(=)g Fy(r)p Fz(h)p FA(\()p Fz(a)p FA(\))p Fz(:)286 b FA(\(5.18\))515 2564 y(So)29 b(w)n(e)h(ha)n(v)n(e)e(got)h(the)h(system)g(\(5.1\),)g (written)g(in)g(the)g(form)g(\(5.8\))o(,)h(with)f Fz(")f FA(replaced)g(b)n(y)515 2663 y Fz(")554 2633 y Fv(1)p Fw(=)p Fv(3)658 2663 y FA(.)35 b(If)21 b(Theorem)41 b(5.1)20 b(applies,)i(then)f(most)g(of)g(the)g(\014nite-gap)f(tori)g Fy(f)p Fz(p)j FA(=)f(const)p Fy(g)e FA(p)r(ersist)515 2763 y(in)33 b(the)g(p)r(erturb)r(ed)g(equation,)h(as)e(states)h(the)g (Metatheorem.)52 b(T)-7 b(o)32 b(b)r(e)i(able)e(to)h(use)g(the)515 2863 y(theorem)27 b(w)n(e)g(ha)n(v)n(e)f(to)i(c)n(hec)n(k)f(the)h (assumptions)e(H1\))i(-)g(H4\).)639 2962 y(The)34 b(condition)g(H2\))g (holds)g(if)h(the)f(in)n(tegrable)f(equation)h(is)g(1D)g(\(if)g(the)h (sp)r(ectrum)515 3062 y(is)30 b(asymptotically)g(double,)i(e.g.)46 b(if)31 b(the)g(unp)r(erturb)r(ed)g(equation)g(is)f(the)h(Sine{Gordon) 515 3162 y(equation)23 b(under)h(the)h(p)r(erio)r(dic)f(b)r(oundary)f (conditions,)i(then)f(one)g(should)g(use)g(a)g(v)n(ersion)515 3261 y(of)36 b(the)g(Metatheorem,)i(based)d(on)h(the)g(Chierc)n(hia{Y) -7 b(ou)35 b(result\).)62 b(The)36 b(quasilinearit)n(y)515 3361 y(condition)27 b(H3\))h(holds)f(due)h(to)f(\(5.17\).)37 b(The)27 b(assumption)g(H1\))h(no)n(w)f(tak)n(es)g(the)h(form)1693 3536 y(Hess)13 b Fz(h)p FA(\()p Fz(p)p FA(\))24 b Fy(6\021)e FA(0)p Fz(:)966 b FA(\(5.19\))515 3711 y(This)22 b(is)h(exactly)f (Kolmogoro)n(v's)d(nondegeneracy)h(condition)j(for)f(the)h(in)n (tegrable)e(system)515 3811 y(on)36 b Fy(T)705 3781 y Fv(2)p Fw(n)784 3811 y FA(.)63 b(The)37 b(assumption)f(H4\))g(with)h Fz(!)k FA(=)c Fy(r)p Fz(h)p FA(\()p Fz(a)p FA(\))g(is)f(the)h(second)f (nondegeneracy)515 3910 y(condition,)27 b(whic)n(h)h(needs)f(v)n (eri\014cation.)639 4010 y(Summing)39 b(up)g(what)g(w)n(as)f(said)g(ab) r(o)n(v)n(e,)i(w)n(e)f(see)f(that)h(Theorem)f(5.1)g(implies)h(the)515 4110 y(Metatheorem)19 b(if)i(the)f(unp)r(erturb)r(ed)h(in)n(tegrable)e (equation)h(is)g(1D)g(quasilinear,)g(the)h(linear)515 4209 y(op)r(erator)i Fz(G)p FA(\()p Fz(p;)28 b(q)s FA(\))e(from)f(Step) g(1)g(p)r(ossesses)f(the)h(required)f(regularit)n(y)g(prop)r(erly)g (and)h(the)515 4309 y(nondegeneracy)g(assumptions)i(\(5.19\))g(and)g (\(5.7\))h(hold)f(true.)639 4408 y(The)i(sc)n(heme)g(w)n(e)f(ha)n(v)n (e)g(just)h(explained)g(w)n(as)f(suggested)f(in)i([Kuk89)o(],)g(where)g (it)g(w)n(as)515 4508 y(used)h(to)h(pro)n(v)n(e)e(an)i(abstract)e (KAM-theorem,)i(whic)n(h)g(next)g(w)n(as)f(applied)g(to)h(Birkho\013-) 515 4608 y(in)n(tegrable)g(in\014nite)j(dimensional)e(systems)g(and)h (to)g(p)r(erturb)r(ed)g(KdV)f(equations.)52 b(See)515 4707 y([Kuk00)n(],)37 b([EKMY02)o(])e(for)f(a)h(more)f(general)f (abstract)h(theorem,)i(based)f(on)f(the)i(same)515 4807 y(sc)n(heme.)639 4907 y(Steps)20 b(1{2)e(are)g(not)h(the)h(only)f(w)n (a)n(y)f(to)h(reduce)g(an)g(in)n(tegrable)f(equation)g(to)h(the)h (normal)515 5006 y(form)g(\(5.16\))o(.)35 b(Another)21 b(approac)n(h)e(to)h(get)h(it)g(had)g(b)r(een)g(initiated)g(b)n(y)g (Kapp)r(eler)f([Kap91)n(].)1905 5255 y(22)p eop PStoPSsaved restore %%Page: (22,23) 12 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 23 22 bop 515 523 a FA(It)20 b(w)n(as)g(dev)n(elop)r(ed)g(further)g(in) h(a)f(n)n(um)n(b)r(er)g(of)g(publications)g(and)g(\014nally)g(in)h ([KM01)o(])f(it)h(w)n(as)515 623 y(pro)n(v)n(ed)c(that)j(the)f(KdV)g (equation)g(is)g(Birkho\013-in)n(tegrable.)31 b(It)20 b(means)e(the)i(follo)n(wing.)33 b(Let)515 722 y(us)c(tak)n(e)g(the)h (Darb)r(oux)f(scale)f(\()p Fy(f)p Fz(X)1640 734 y Fw(s)1676 722 y Fy(g)p Fz(;)14 b(\013)1808 734 y Fv(2)1844 722 y FA(\))30 b(with)g(the)g(index-set)f Fy(Z)k FA(=)26 b Fx(Z)2842 734 y Fv(0)2874 722 y FA(,)k(and)f Fz(\022)3129 734 y Fw(k)3196 722 y FA(=)d Fy(j)p Fz(k)s Fy(j)515 822 y FA(\(see)21 b(Example)42 b(2.6\).)35 b(Then)21 b(there)g(exists)h(a)f (map)g(\010)i(:)g Fz(X)2311 834 y Fu(1)2404 822 y Fy(!)g Fz(H)2586 792 y Fu(1)2656 822 y FA(\()p Fz(S)2744 792 y Fv(1)2782 822 y FA(\))2814 834 y Fv(0)2873 822 y FA(whic)n(h)e (extends)515 922 y(to)27 b(analytic)g(maps)g Fz(X)1218 934 y Fw(s)1277 922 y Fy(!)c Fz(H)1459 891 y Fw(s)1494 922 y FA(\()p Fz(S)1582 891 y Fv(1)1619 922 y FA(\))1651 934 y Fv(0)1689 922 y FA(,)28 b Fz(s)23 b Fy(\025)f FA(0,)27 b(suc)n(h)h(that)613 1162 y Fz(h)19 b Fy(\016)e FA(\010\()p Fz(u)p FA(\))24 b(=)1049 1058 y Fu(1)1022 1083 y Fq(X)1025 1260 y Fw(j)s Fv(=1)1156 1162 y Fz(j)1195 1128 y Fv(3)1232 1162 y FA(\()p Fz(u)1312 1128 y Fv(2)1312 1183 y Fw(j)1368 1162 y FA(+)18 b Fz(u)1499 1128 y Fv(2)1499 1183 y Fu(\000)p Fw(j)1585 1162 y FA(\))h(+)f Fy(h)p FA(a)27 b(function)h(of)48 b Fz(u)2308 1128 y Fv(2)2308 1183 y Fw(l)2364 1162 y FA(+)18 b Fz(u)2495 1128 y Fv(2)2495 1183 y Fu(\000)p Fw(l)2571 1162 y Fz(;)28 b(l)d FA(=)d(1)p Fz(;)14 b FA(2)p Fz(;)g(:)g(:)g(:)o Fy(i)p Fz(:)99 b FA(\(5.20\))515 1425 y(Here)25 b Fy(f)p Fz(u)799 1437 y Fw(k)839 1425 y Fz(;)j(k)e Fy(2)d Fx(Z)1098 1437 y Fv(0)1130 1425 y Fy(g)i FA(are)g(co)r (e\016cien)n(ts)g(of)h(decomp)r(osition)f(of)h Fz(u)d Fy(2)g Fz(X)2701 1437 y Fw(s)2762 1425 y FA(in)k(the)f(basis)f Fy(f)p Fz(')3297 1437 y Fw(k)3337 1425 y Fy(g)515 1524 y FA(and)32 b Fz(h)h FA(is)g(the)g(KdV-hamiltonian)g(\(see)f(Example)g (2.7\).)53 b(Moreo)n(v)n(er,)31 b(the)i(hamiltonian)515 1624 y(\(5.20\))28 b(de\014nes)g(an)g(analytic)g(Hamiltonian)h(v)n (ector)e(\014eld)i(of)g(order)e(three)h(in)h(eac)n(h)f(space)515 1724 y Fz(X)584 1736 y Fw(d)622 1724 y FA(,)34 b Fz(d)d Fy(\025)g FA(1.)51 b(In)33 b(the)g(transformed)e(v)-5 b(ariables)31 b(the)i Fz(N)9 b FA(-gap)31 b(tori)h(of)g(the)h(KdV)g (equation)515 1823 y(tak)n(e)k(the)i(form)e(\(5.5\),)k(where)c Fz(n)k Fy(\025)f Fz(N)47 b FA(and)38 b(exactly)f Fz(N)47 b FA(n)n(um)n(b)r(ers)38 b Fz(I)2824 1835 y Fw(j)2898 1823 y FA(are)f(non-zero.)515 1923 y(No)n(w)e(let)h(us)g(tak)n(e)f(a)g (torus)g(\(5.5\),)j(where)d Fz(I)43 b Fy(2)37 b Fx(R)2146 1893 y Fw(n)2146 1943 y Fv(+)2207 1923 y FA(.)62 b(Making)35 b(a)g(c)n(hange)g(of)g(v)-5 b(ariables)515 2023 y(as)34 b(in)g(section)h(5.1,)g(w)n(e)f(arriv)n(e)f(at)h(the)h(hamiltonian)f (\(5.18\))o(.)58 b(Detailed)35 b(and)f(readable)515 2122 y(deriv)-5 b(ation)27 b(of)g(the)h(normal)f(form)g(\(5.20\))g(see)g(in) h([KP03)n(].)639 2222 y(Reduction)38 b(to)f(the)g(Birkho\013)g(normal)f (form)h(\(5.20\))f(uses)h(essen)n(tially)f(sp)r(eci\014cs)h(of)515 2321 y(the)e(KdV's)g Fz(L)p FA(-op)r(erator.)57 b(Still,)38 b(similar)c(argumen)n(ts)g(apply)h(as)f(w)n(ell)h(to)g(the)g(defo)r (cus-)515 2421 y(ing)e(Zakharo)n(v{Shabat)c(equation,)34 b(see)f([GK03)o(].)53 b(Presumably)-7 b(,)33 b(the)h(Birkho\013)e (normal)515 2521 y(forms)i(exist)h(for)f(some)h(other)f(in)n(tegrable)g (equations)g(with)h(self-adjoin)n(t)g Fz(L)p FA(-op)r(erators,)515 2620 y(but)24 b(not)g(for)g(equations)f(with)h(non-selfadjoin)n(t)g(op) r(erators.)33 b(In)25 b(particular,)e(the)h(fo)r(cusing)515 2720 y(Zakharo)n(v{Shabat)i(equation)j(cannot)g(b)r(e)i(reduced)e(to)h (the)g(form)f(\(5.20\))g(since)h(for)f(this)515 2820 y(equation)24 b(some)h(\014nite-gap)g(tori)g(are)f(linearly)g(unstable) i([CMM02)o(],)g(while)f(all)g(in)n(v)-5 b(arian)n(t)515 2919 y(tori)27 b(of)g(the)h(form)g(\(5.5\))f(for)g(the)h(hamiltonian)f (\(5.20\))g(are)g(linearly)f(stable.)515 3049 y Fp(Example)37 b FA(5.7)28 b Fp(\(p)l(erturb)l(e)l(d)i(KdV)g(e)l(quation\).)42 b FA(Consider)26 b(the)i(equation)744 3266 y(_)-38 b Fz(u)p FA(\()p Fz(t;)28 b(x)p FA(\))23 b(=)1090 3210 y(1)p 1090 3247 42 4 v 1090 3323 a(4)1175 3210 y Fz(@)p 1151 3247 97 4 v 1151 3323 a(@)5 b(x)1258 3266 y FA(\()p Fz(u)1338 3232 y Fu(00)1398 3266 y FA(+)18 b(3)p Fz(u)1571 3232 y Fv(2)1626 3266 y FA(+)g Fz("f)9 b FA(\()p Fz(x;)28 b(u)p FA(\)\))p Fz(;)97 b(x)23 b Fy(2)h Fz(S)2365 3232 y Fv(1)2402 3266 y FA(;)2485 3153 y Fq(Z)2531 3342 y Fw(S)2575 3325 y Ft(1)2625 3266 y Fz(u)14 b(dx)23 b Fy(\021)g FA(0)p Fz(;)214 b FA(\(5.21\))515 3482 y(where)35 b Fz(f)45 b FA(is)36 b(smo)r(oth)g(in)g Fz(x;)28 b(u)36 b FA(and)g(analytic)f(in) h Fz(u)p FA(.)62 b(The)36 b(Metatheorem)g(applies)f(and)515 3582 y(implies)30 b(that)h(most)f(of)g(\014nite-gap)f(KdV-solutions)g (p)r(ersist)h(as)g(time-quasip)r(erio)r(dic)f(so-)515 3681 y(lutions)e(of)34 b(\(5.21\).)i(Moreo)n(v)n(er,)25 b(these)j(solutions)f(are)f(smo)r(oth)i(and)f(linearly)g(stable.)639 3781 y(This)d(result)f(w)n(as)g(\014rst)g(pro)n(v)n(ed)f(in)i([Kuk89)n (])g(with)g(a)f(n)n(um)n(b)r(er)g(of)h(omissions.)34 b(Tw)n(o)23 b(the)515 3881 y(most)30 b(serious)f(are)h(that)h(Theorem)e (5.1,)i(pro)n(v)n(ed)e(then)i(only)f(for)g(semilinear)g(equations,)515 3980 y(w)n(as)24 b(used)h(in)h(a)f(quasilinear)f(case,)h(and)g(that)g (the)h(non-degeneracy)d(assumptions)i(\(5.19\))515 4080 y(and)37 b(\(5.7\))g(w)n(ere)f(tak)n(en)h(for)f(gran)n(ted.)65 b(These)37 b(omissions)f(w)n(ere)g(\014lled)h(in)h(later.)65 b(The)515 4179 y(quasilinear)20 b(v)n(ersion)h(of)g(Theorem)h(5.1)f(w)n (as)g(pro)n(v)n(ed)f(in)i([Kuk98)o(])g(\(preprin)n(t)f(of)h(this)h(pap) r(er)515 4279 y(app)r(eared)28 b(in)h(1995\),)f(and)h(the)h (non-degeneracy)d(conditions)h(w)n(ere)h(v)n(eri\014ed)f(in)h([BK91)o (].)515 4379 y(Also)j(see)g([Kuk00)n(],)i(section)e(6.2.1.)50 b(The)32 b(argumen)n(ts)f(in)i([BK91)o(,)f(Kuk00)o(])g(are)f(general) 515 4478 y(and)c(applies)g(to)h(other)f(equations.)639 4578 y(F)-7 b(or)26 b(a)f(complete)h(pro)r(of)g(of)g(`KAM)g(for)g(KdV') g(see)f([Kuk00)o(,)h(EKMY02)o(])g(and)g([KP03)n(].)p 3318 4678 4 57 v 3322 4625 50 4 v 3322 4678 V 3372 4678 4 57 v 639 4807 a(The)21 b(Metatheorem)g(\(in)g(its)h(rigorous)c(form)j (as)g(in)g([Kuk00)o(,)g(EKMY02)n(])g(and)g([KP03)o(]\),)515 4907 y(applies)33 b(to)h(quasilinear)f(Hamiltonian)g(p)r(erturbations)h (of)g(an)n(y)f(higher)g(equation)g(from)515 5006 y(the)c(KdV-hierarc)n (h)n(y)-7 b(,)26 b(pro)n(vided)i(that)h(the)g(non-degeneracy)d (relations)i(are)f(c)n(hec)n(k)n(ed)h(for)1905 5255 y(23)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 24 23 bop 515 523 a FA(this)29 b(equation.)41 b(It)30 b(can)f(b)r(e)g(done)g(in)h(the)g(same)e(w)n(a)n(y)g(as)h(in)g(Example) g(5.7.)40 b(See)30 b([KP03)n(],)515 623 y(where)d(the)h(nondegeneracy)d (of)j(the)g(second)f(KdV)h(equation)f(is)g(v)n(eri\014ed.)515 756 y Fp(Example)37 b FA(5.8)28 b Fp(\(p)l(erturb)l(e)l(d)i(SG)g(e)l (quation\).)42 b FA(Consider)26 b(the)i(equation)1036 938 y(\177)-48 b Fz(u)23 b FA(=)f Fz(u)1236 950 y Fw(xx)1334 938 y Fy(\000)c FA(sin)c Fz(u)k FA(+)g Fz("f)9 b FA(\()p Fz(u;)26 b(x)p FA(\))p Fz(;)98 b(u)p FA(\()p Fz(t;)28 b FA(0\))23 b(=)f Fz(u)p FA(\()p Fz(t;)28 b(\031)s FA(\))23 b(=)g(0)p Fz(;)303 b FA(\(5.22\))515 1121 y(where)33 b Fz(f)9 b FA(\(0)p Fz(;)27 b(x)p FA(\))34 b Fy(\021)f FA(0)h(\(and)g Fz(f)42 b Fy(2)34 b Fz(C)1659 1091 y Fu(1)1763 1121 y FA(is)g(analytic)f(in)h Fz(u)p FA(\).)55 b(The)34 b(Metatheorem)f(applies)515 1220 y(to)25 b(pro)n(v)n(e)f(p)r (ersistence)h(most)g(of)g(\014nite-gap)g(solutions)g(of)g(the)h (SG-equation,)f(see)g([BK93)o(,)515 1320 y(Kuk00)n(,)33 b(EKMY02)o(].)53 b(In)33 b(general,)h(due)f(to)g(the)g(phenomenon)g (explained)g(in)g(Example)515 1420 y(2.9,)26 b(the)h(p)r(ersisted)g (solutions)e(are)h(only)h Fz(H)1903 1390 y Fv(2)1940 1420 y FA(-smo)r(oth)f(in)h Fz(x)p FA(.)37 b(But)27 b(if)g Fz(f)35 b FA(is)27 b Fz(x)p FA(-indep)r(enden)n(t)515 1519 y(and)g(o)r(dd)h(in)g Fz(u)p FA(,)f(then)h(they)g(are)f(smo)r (oth.)639 1619 y(In)f(di\013erence)g(with)h(the)f(KdV-case,)f(large)g (amplitude)h(\014nite-gap)f(SG-solutions,)h(as)515 1719 y(w)n(ell)20 b(as)g(the)h(corresp)r(onding)e(p)r(ersisted)h(solutions)g (of)27 b(\(5.22\))o(,)c(in)d(general)g(are)f(not)i(linearly)515 1818 y(stable.)p 3318 1818 4 57 v 3322 1765 50 4 v 3322 1818 V 3372 1818 4 57 v 639 2001 a(T)-7 b(o)26 b(end)g(this)h(section)e (w)n(e)h(note)g(that)g(since)g(the)h(p)r(ersisted)f(solutions)f Fz(u)2918 2013 y Fw(")2953 2001 y FA(\()p Fz(t)p FA(\))i(ha)n(v)n(e)e (the)515 2100 y(form)948 2200 y Fz(u)996 2212 y Fw(")1031 2200 y FA(\()p Fz(t)p FA(\))e(=)g(\010)1296 2212 y Fw(")1331 2200 y FA(\()p Fz(p;)28 b(q)22 b FA(+)c Fz(t!)1680 2212 y Fw(")1715 2200 y FA(\()p Fz(p)p FA(\)\))24 b(=)e(\010)2024 2212 y Fv(0)2061 2200 y FA(\()p Fz(p;)28 b(q)22 b FA(+)c Fz(t!)2410 2212 y Fw(")2445 2200 y FA(\()p Fz(p)p FA(\)\))h(+)f Fz(O)r FA(\()2782 2136 y Fy(p)p 2852 2136 39 4 v 2852 2200 a Fz(")p FA(\))p Fz(;)515 2350 y FA(then)35 b(to)f(calculate)g (them)h(with)f(the)h(accuracy)2087 2290 y Fy(p)p 2156 2290 V 60 x Fz(")f FA(for)g(all)g(v)-5 b(alues)34 b(of)g(time)h Fz(t)p FA(,)h(w)n(e)e(can)515 2449 y(use)h(the)i(\\\014nite)e(gap)g (map")h(\010)1539 2461 y Fv(0)1612 2449 y FA(with)g(the)g(corrected)e (frequency)i(v)n(ector.)60 b(Moreo)n(v)n(er,)515 2549 y Fz(!)567 2561 y Fw(")602 2549 y FA(\()p Fz(p)p FA(\))28 b(=)f Fy(r)p Fz(h)p FA(\()p Fz(p)p FA(\))21 b(+)f Fz("W)1274 2561 y Fv(1)1311 2549 y FA(\()p Fz(p)p FA(\))h(+)f Fz(O)r FA(\()p Fz(")1659 2519 y Fv(2)1697 2549 y FA(\),)31 b(where)f(the)h(v)n (ector)e Fz(W)2502 2561 y Fv(1)2540 2549 y FA(\()p Fz(p)p FA(\))h(can)g(b)r(e)h(obtained)f(b)n(y)515 2648 y(a)n(v)n(eraging)25 b(o)n(v)n(er)h(the)j(corresp)r(onding)d(\014nite-gap)i(torus)f(of)h (some)g(explicit)g(quan)n(tit)n(y)-7 b(,)28 b(see)515 2748 y([Kuk00)n(],)g(p.147.)515 2980 y Fs(5.6)112 b(Small)36 b(amplitude)g(solutions)g(of)i(HPDEs)515 3134 y FA(Let)27 b(us)h(consider)f(the)h(nonlinear)e(string)h(equation)556 3316 y Fz(u)604 3328 y Fw(tt)681 3316 y FA(=)c Fz(u)817 3328 y Fw(xx)904 3316 y Fy(\000)8 b Fz(mu)g FA(+)g Fz(f)h FA(\()p Fz(u)p FA(\))p Fz(;)59 b(u)22 b FA(=)h Fz(u)p FA(\()p Fz(t;)28 b(x)p FA(\))p Fz(;)37 b FA(0)23 b Fy(\024)g Fz(x)g Fy(\024)g Fz(\031)s FA(;)97 b Fz(u)p FA(\()p Fz(t;)27 b FA(0\))c(=)g Fz(u)p FA(\()p Fz(t;)k(\031)s FA(\))d(=)f(0)p Fz(:)41 b FA(\(5.23\))515 3499 y(Here)27 b Fz(m)c(>)g FA(0)k(and)g Fz(f)36 b FA(is)28 b(an)f(o)r(dd)h(analytic)f(function)h (of)g(the)g(form)1415 3682 y Fz(f)9 b FA(\()p Fz(u)p FA(\))22 b(=)h Fz(\024u)1783 3647 y Fv(3)1838 3682 y FA(+)18 b Fz(O)r FA(\()p Fz(u)2066 3647 y Fv(5)2104 3682 y FA(\))p Fz(;)97 b(\024)23 b(>)g FA(0)p Fz(:)515 3864 y FA(Since)28 b Fz(m;)f(\024)d(>)f FA(0,)k(then)i(constan)n(ts)e Fz(a;)g(b)c(>)g FA(0)28 b(can)f(b)r(e)i(found)f(suc)n(h)f(that)h Fy(\000)p Fz(mu)18 b FA(+)g Fz(f)9 b FA(\()p Fz(u)p FA(\))23 b(=)515 3964 y Fy(\000)p Fz(a)14 b FA(sin)f Fz(bu)p FA(.)36 b(Hence,)28 b(the)g(equation)f(\(5.23\))g(can)g(b)r(e)h(written)g(as) 1405 4147 y Fz(u)1453 4159 y Fw(tt)1530 4147 y FA(=)23 b Fz(u)1666 4159 y Fw(xx)1763 4147 y Fy(\000)18 b Fz(a)c FA(sin)g Fz(bu)j FA(+)h Fz(O)r FA(\()p Fy(j)p Fz(u)p Fy(j)2395 4112 y Fv(5)2433 4147 y FA(\))p Fz(:)515 4329 y FA(After)32 b(the)h(scaling)e Fz(u)f FA(=)g Fz("w)r FA(,)k Fz(")c Fy(\034)h FA(1,)i(the)f(higher-order)e(p)r(erturbation)h (transforms)g(to)515 4429 y(a)k(small)g(one,)j(and)d(w)n(e)g(can)g (apply)h(the)g(Metatheorem)f(\(cf.)61 b(Example)35 b(5.8\))g(to)h(pro)n (v)n(e)515 4528 y(that)24 b(small-amplitude)f(parts)g(of)h(the)g (\014nite-gap)f(manifolds)h Fy(T)2539 4498 y Fv(2)p Fw(n)2617 4528 y FA(,)h Fz(n)e FA(=)f(1)p Fz(;)14 b FA(2)p Fz(;)g(:)g(:)g(:)f FA(,)24 b(for)g(the)515 4628 y(SG)h(equation)g Fz(u)1036 4640 y Fw(tt)1113 4628 y FA(=)e Fz(u)1249 4640 y Fw(xx)1342 4628 y Fy(\000)13 b Fz(a)h FA(sin)f Fz(bu)25 b FA(with)h(the)g(Diric)n (hlet)f(b)r(oundary)g(conditions)f(mostly)515 4728 y(p)r(ersist)j(in)h (\(5.23\))o(.)37 b(T)-7 b(o)27 b(put)i(this)e(sc)n(heme)h(through,)e (the)i(small-amplitude)g(parts)1012 4910 y Fy(T)1079 4876 y Fv(2)p Fw(n)1057 4931 y(\016)1180 4910 y FA(=)23 b Fy(f)p FA(\()p Fz(u;)k FA(_)-36 b Fz(u)n FA(\))24 b Fy(2)f(T)1673 4876 y Fv(2)p Fw(n)1775 4910 y Fy(j)g(k)p Fz(u)p Fy(k)17 b FA(+)h Fy(k)c FA(_)-37 b Fz(u)o Fy(k)22 b Fz(<)h(\016)s Fy(g)p Fz(;)97 b FA(0)22 b Fz(<)h(\016)j Fy(\034)d FA(1)p Fz(;)1905 5255 y FA(24)p eop PStoPSsaved restore %%Page: (24,25) 13 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 25 24 bop 515 523 a FA(of)26 b(the)g(manifolds)g Fy(T)1190 493 y Fv(2)p Fw(n)1295 523 y FA(ha)n(v)n(e)f(to)h(b)r(e)g(studied)h(in) f(details.)36 b(This)26 b(task)g(w)n(as)f(accomplished)515 623 y(in)j([BK95b)n(],)g(where)f(the)h(follo)n(wing)f(results)g(w)n (ere)f(pro)n(v)n(ed:)639 734 y(i\))i(the)g(sets)p 1028 658 145 4 v 28 w Fy(T)1094 706 y Fv(2)p Fw(n)1073 759 y(\016)1200 734 y FA(are)f(smo)r(oth)g(manifolds)g(whic)n(h)h(con)n (tain)f(the)h(origin,)639 834 y(ii\))34 b(they)g(are)e(in)i(one-to-one) e(corresp)r(ondence)f(with)j(their)g(tangen)n(t)f(spaces)f(at)i(the)515 934 y(origin,)639 1033 y(iii\))22 b(these)g(tangen)n(t)f(spaces)f(are)h (in)n(v)-5 b(arian)n(t)20 b(spaces)h(for)g(the)h(Klein)f({)g(Gordon)g (equation)515 1133 y Fz(u)563 1145 y Fw(tt)640 1133 y FA(=)h Fz(u)775 1145 y Fw(xx)873 1133 y Fy(\000)c FA(\()p Fz(ab)p FA(\))p Fz(u)p FA(.)639 1233 y(Another)28 b(pro)r(of)g(of)g (i\)-iii\))g(w)n(as)g(suggested)f(in)h([Kuk00)o(].)38 b(It)29 b(is)f(based)f(on)h(some)g(ideas)515 1332 y(from)18 b([Kap91)n(])h(and)g(applies)f(to)h(other)f(in)n(tegrable)f(equations.) 33 b(After)19 b(i\)-iii\))h(are)d(obtained,)515 1432 y(a)34 b(v)n(ersion)f(of)i(the)g(Metatheorem)g(\(or)f(a)g(v)n(ersion)g (of)g(Theorem)g(5.1\))h(applies)f(to)h(pro)n(v)n(e)515 1531 y(that)e(most)g(of)h(\014nite-gap)e(solutions)h(from)g(a)g (manifold)g Fy(T)2429 1501 y Fv(2)p Fw(n)2408 1555 y(\016)2541 1531 y FA(p)r(ersist)g(in)h(\(5.23\))e(in)i(the)515 1631 y(follo)n(wing)27 b(sense:)39 b(the)29 b(2)p Fz(n)p FA(-dimensional)f (Hausdor\013)g(measure)f(of)i(the)g(p)r(ersisted)g(part)f(of)515 1731 y(the)h(manifold,)h(divided)g(b)n(y)f(a)g(similar)g(measure)f(of)h Fy(T)2274 1701 y Fv(2)p Fw(n)2252 1754 y(\016)2352 1731 y FA(,)h(con)n(v)n(erges)c(to)k(one)f(as)f Fz(\016)h Fy(!)d FA(0.)515 1830 y(See)h([BK95a)n(])h(for)f(a)g(pro)r(of)g(and)h ([Kuk94)n(])g(for)f(discussion.)639 1930 y(Similar)g(results)h(hold)f (for)g(the)h(NLS)g(equation)980 2113 y Fz(i)14 b FA(_)-37 b Fz(u)22 b FA(=)g Fz(u)1214 2125 y Fw(xx)1312 2113 y FA(+)c Fz(mu)g FA(+)g Fz(f)9 b FA(\()p Fy(j)p Fz(u)p Fy(j)1793 2078 y Fv(2)1829 2113 y FA(\))p Fz(u;)97 b(f)9 b FA(\(0\))23 b(=)g(0)p Fz(;)41 b(f)2452 2078 y Fu(0)2475 2113 y FA(\(0\))23 b(=)f Fz(\015)28 b Fy(6)p FA(=)23 b(0)p Fz(;)252 b FA(\(5.24\))515 2295 y(where)36 b Fz(f)45 b FA(is)36 b(analytic,)i(since)f(it)g(is)f(a)g(higher-order)e(p)r (erturbation)i(of)g(the)h(Zakharo)n(v{)515 2395 y(Shabat)29 b(equation)g(\(4.7\).)42 b(But)30 b(it)g(turns)g(out)f(that)h(it)g(is)f (easier)g(to)g(appro)n(ximate)f(\(5.24\))515 2494 y(near)g(the)h (origin)e(b)n(y)i(its)g(partial)e(Birkho\013)i(normal)e(form.)40 b(The)29 b(latter)f(is)h(an)f(in)n(tegrable)515 2594 y(in\014nite-dimensional)c(Hamiltonian)g(system)h(\(whic)n(h)f(is)h (not)f(an)g(HPDE\),)h(and)f(a)g(sibling)515 2694 y(of)19 b(the)g(Metatheorem)g(applies)f(to)h(pro)n(v)n(e)f(that)h(most)g(of)g (its)g(time-quasip)r(erio)r(dic)g(solutions)515 2793 y(p)r(ersist)h(in)h(\(5.24\),)h(see)e([KP96)o(].)34 b(More)20 b(on)h(the)g(tec)n(hniques)g(of)g(Birkho\013)f(normal)g(forms)g(in)515 2893 y(HPDE)i(see)h(in)f([P\177)-42 b(os96b)n(])23 b(and)g([KP03)n(].) 35 b(The)23 b(classical)e(reference)h(for)g(\014nite-dimensional)515 2993 y(Birkho\013)27 b(normal)f(forms)h(is)h(the)g(b)r(o)r(ok)f([MS71)o (].)515 3267 y FC(6)134 b(Around)44 b(the)h(Nekhoroshev)h(theorem)515 3449 y FA(The)25 b(classical)e(Nekhoroshev)h(theorem)g([Nek77)o(])h (deals)f(with)i(nearly-in)n(tegrable)c(Hamil-)515 3549 y(tonian)32 b(systems)h(with)g(analytic)f(hamiltonians)g Fz(H)2175 3561 y Fw(")2211 3549 y FA(\()p Fz(p;)c(q)s FA(\))k(=)f Fz(h)p FA(\()p Fz(p)p FA(\))22 b(+)g Fz("H)7 b FA(\()p Fz(p;)27 b(q)s FA(\))33 b(on)g(the)515 3648 y(phase-space)18 b Fz(P)d Fy(\002)s Fx(T)1152 3618 y Fw(n)1196 3648 y FA(,)21 b Fz(P)35 b Fy(\032)23 b Fx(R)1470 3618 y Fw(n)1521 3648 y FA(,)e(giv)n(en)e(the)i(usual)e(symplectic)h (structure)f Fz(dp)s Fy(^)s Fz(dq)s FA(.)35 b(Under)515 3748 y(the)29 b(assumption)f(that)h(the)g(hamiltonian)g Fz(h)p FA(\()p Fz(p)p FA(\))g(satis\014es)f(a)g(mild)h(non-degeneracy)e (con-)515 3847 y(dition)h(called)g Fp(the)i(ste)l(epness)p FA(,)f(the)f(theorem)g(states)g(that)g(the)h(action)e(v)-5 b(ariables)27 b(c)n(hange)515 3947 y(exp)r(onen)n(tially)34 b(slo)n(w)f(along)g(tra)5 b(jectories)33 b(of)i(the)g(system.)57 b(Namely)-7 b(,)36 b(there)e(exist)h(con-)515 4047 y(stan)n(ts)d Fz(a;)27 b(b)k Fy(2)h FA(\(0)p Fz(;)27 b FA(1\))33 b(suc)n(h)f(that)h (for)f(an)n(y)g(tra)5 b(jectory)31 b(\()p Fz(p)p FA(\()p Fz(t)p FA(\)\))p Fz(;)d(q)s FA(\()p Fz(t)p FA(\)\))34 b(of)e(the)h(system)f(w)n(e)515 4146 y(ha)n(v)n(e)1267 4246 y Fy(j)p Fz(p)p FA(\()p Fz(t)p FA(\))19 b Fy(\000)f Fz(p)p FA(\(0\))p Fy(j)23 b(\024)g Fz(C)6 b(")1914 4212 y Fw(a)2000 4246 y FA(if)53 b Fy(j)p Fz(t)p Fy(j)23 b(\024)g FA(exp\()p Fz(")2486 4212 y Fu(\000)p Fw(b)2571 4246 y FA(\))p Fz(:)582 b FA(\(6.1\))515 4395 y(Strictly)30 b(con)n(v)n(ex)f(functions)i Fz(h)p FA(\()p Fz(p)p FA(\))g(form)f(an)h (imp)r(ortan)n(t)f(class)f(of)i(the)g(steep)f(hamiltoni-)515 4495 y(ans.)40 b(An)30 b(alternativ)n(e)d(pro)r(of)i(of)g(the)g (theorem)f(whic)n(h)h(applies)g(in)g(the)g(con)n(v)n(ex)f(case)g(w)n (as)515 4595 y(suggested)i(b)n(y)h(Lo)r(c)n(hak)f([Lo)r(c92)o(].)48 b(It)32 b(is)f(based)g(on)g(clev)n(er)f(appro)n(ximation)g(of)h(a)g (tra)5 b(jec-)515 4694 y(tory)20 b(\()p Fz(p)p FA(\()p Fz(t)p FA(\))p Fz(;)29 b(q)s FA(\()p Fz(t)p FA(\)\))22 b(b)n(y)f(a)g(time-p)r(erio)r(dic)h(solution)e(of)i(the)g(equation)e (whic)n(h)i(is)f(a)g(high-order)515 4794 y(normal)g(form)g(for)h Fz(H)1172 4806 y Fw(")1208 4794 y FA(.)35 b(So)21 b(rational)g (frequency-v)n(ectors)f(pla)n(y)h(for)h(the)g(Lo)r(c)n(hak)f(approac)n (h)515 4894 y(v)n(ery)26 b(imp)r(ortan)n(t)h(role.)1905 5255 y(25)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 26 25 bop 639 523 a FA(Original)20 b(Nekhoroshev's)g(pro)r(of)h(con)n (tains)g(t)n(w)n(o)g(parts,)h(analytical)e(and)i(geometrical.)515 623 y(The)30 b(tec)n(hniques,)g(dev)n(elop)r(ed)f(in)i(the)f (analytical)f(part)g(of)h(the)g(pro)r(of,)g(allo)n(w)f(to)h(get)g(the) 515 722 y(follo)n(wing)j(result,)k(whic)n(h)d(w)n(e)g(call)h(b)r(elo)n (w)f(the)h(quasi-Nekhoroshev)d(theorem:)50 b(Let)35 b(us)515 822 y(consider)24 b(the)i(hamiltonian)f Fz(H)1505 834 y Fw(")1540 822 y FA(,)h(dep)r(ending)g(on)f(an)g(additional)g(v)n (ector-parameter)d Fz(!)k Fy(2)515 922 y FA(\012)41 b Fe(b)g Fx(R)776 891 y Fw(n)827 922 y FA(,)g Fz(H)960 934 y Fw(")1037 922 y FA(=)g Fz(p)26 b Fy(\001)f Fz(!)k FA(+)c Fz("H)7 b FA(\()p Fz(p;)27 b(q)s FA(\).)70 b(Then)39 b(for)f(an)n(y)g Fz(\015)46 b(>)40 b FA(0)f(there)f(exists)g(a)g(Borel) 515 1021 y(subset)28 b(\012)830 1033 y Fw(\015)897 1021 y Fy(\032)c FA(\012)29 b(\(`the)g(Diophan)n(tine)f(subset'\))h(suc)n(h) f(that)h(mes\(\012)p Fy(n)p FA(\012)2759 1033 y Fw(\015)2801 1021 y FA(\))c Fz(<)f(\015)5 b FA(,)28 b(and)g(\(6.1\))515 1121 y(with)33 b Fz(C)k FA(=)30 b Fz(C)959 1133 y Fw(\015)1034 1121 y FA(holds)i(if)h Fz(!)h Fy(2)d FA(\012)1569 1133 y Fw(\015)1611 1121 y FA(.)52 b(Note)32 b(that)g(in)h(the)g(Cartesian)e (co)r(ordinates)f(\()p Fz(x;)f(y)s FA(\),)515 1220 y(corresp)r(onding)g (to)h(the)h(action-angle)e(v)-5 b(ariables)30 b(\()p Fz(p;)e(q)s FA(\))j(\(i.e.,)h Fz(x)2587 1232 y Fw(j)2651 1220 y FA(=)2744 1155 y Fq(p)p 2827 1155 119 4 v 65 x FA(2)p Fz(p)2911 1232 y Fw(j)2959 1220 y FA(cos)13 b Fz(q)3121 1232 y Fw(j)3156 1220 y FA(,)31 b Fz(y)3251 1232 y Fw(j)3314 1220 y FA(=)515 1267 y Fq(p)p 598 1267 V 65 x FA(2)p Fz(p)682 1344 y Fw(j)730 1332 y FA(sin)14 b Fz(q)883 1344 y Fw(j)918 1332 y FA(\),)28 b(the)g(hamiltonian)f Fz(H)1674 1344 y Fw(")1737 1332 y FA(reeds)g(as)1303 1564 y Fz(H)1372 1576 y Fw(")1430 1564 y FA(=)1528 1508 y(1)p 1528 1545 42 4 v 1528 1621 a(2)1593 1485 y Fq(X)1727 1564 y Fz(!)1779 1576 y Fw(j)1814 1564 y FA(\()p Fz(x)1893 1530 y Fv(2)1893 1584 y Fw(j)1949 1564 y FA(+)18 b Fz(y)2076 1530 y Fv(2)2073 1584 y Fw(j)2113 1564 y FA(\))h(+)f Fz("H)7 b FA(\()p Fz(x;)28 b(y)s FA(\))p Fz(:)515 1770 y FA(That)k(is,)h Fz(H)908 1782 y Fw(")976 1770 y FA(is)f(a)f(p)r (erturbation)h(of)g(the)g(quadratic)f(hamiltonian)h Fz(H)2786 1782 y Fv(0)2823 1770 y FA(.)51 b(So)32 b(the)g(quasi-)515 1869 y(Nekhoroshev)26 b(theorem)i(implies)h(long-time)e(stabilit)n(y)h (of)g(the)h(zero)e(equilibrium)i(for)e(an)515 1969 y(analytical)f (hamiltonian)1262 2152 y Fz(H)7 b FA(\()p Fz(x;)28 b(y)s FA(\))23 b(=)g Fz(H)1724 2164 y Fv(0)1779 2152 y FA(+)18 b Fz(h;)97 b(h)23 b FA(=)g Fz(O)r FA(\()p Fy(j)p FA(\()p Fz(x;)29 b(y)s FA(\))p Fy(j)2539 2117 y Fv(3)2577 2152 y FA(\))p Fz(;)576 b FA(\(6.2\))515 2334 y(pro)n(vided)29 b(that)i(the)h(v)n(ector)d Fz(!)34 b FA(b)r(elongs)c(to)g(the)h (Diophan)n(tine)g(set.)46 b(In)31 b([Nie98])g(Nieder-)515 2434 y(man)26 b(used)g(the)h(Lo)r(c)n(hak)f(approac)n(h)e(to)i(get)h(a) f(stronger)e(theorem)i(on)g(stabilit)n(y)g(for)g(\(6.2\).)515 2534 y(Namely)-7 b(,)28 b(he)g(pro)n(v)n(ed)f(that)h(the)g(equilibrium) h(is)f(stable)f(during)h(the)g(exp)r(onen)n(tially)g(long)515 2633 y(time)e(if)g(the)f(v)n(ector)f Fz(!)k FA(do)r(es)d(not)h (satis\014es)e(resonan)n(t)g(relations)g(up)i(to)f(order)f(four,)i(and) f Fz(h)515 2733 y FA(is)i(con)n(v)n(ex)f(in)i(a)f(certain)g(sense.)1559 2703 y Fv(8)639 2832 y FA(T)-7 b(o)37 b(get)h(a)f(corresp)r(onding)f (theorem)h(whic)n(h)g(applies)g(to)h(all)f(small)g(initial)h(data)f(is) 515 2932 y(a)32 b(non)n(trivial)h(task,)h(resolv)n(ed)d(b)n(y)i (Niederman)g([Nie98)o(])g(b)n(y)g(means)g(of)g(the)h(Lo)r(c)n(hak)e (ap-)515 3032 y(proac)n(h.)639 3181 y(No)38 b(analogy)d(of)j(the)g (Nekhoroshev)d(theorem)i(for)g(HPDEs)g(is)h(kno)n(wn)f(y)n(et,)i(but)f (a)515 3281 y(n)n(um)n(b)r(er)26 b(of)g Fp(ad)j(ho)l(c)i FA(quasi-Nekhoroshev)24 b(theorems)h(for)h(HPDEs)f(w)n(ere)g(pro)n(v)n (ed,)g(mostly)515 3380 y(b)n(y)18 b(Bourgain)f(and)h(Bam)n(busi,)i(see) e([Bam99a)n(,)h(Bam99b)n(,)g(Bou00)o(])f(and)h(references)e(therein.) 515 3480 y(These)32 b(w)n(orks)f(discuss)h(stabilit)n(y)g(of)g(the)h (equilibrium)g(for)e(HPDEs)h(\(mostly)h(1D\))g(with)515 3580 y(hamiltonians)18 b(of)h(the)g(form)f(\(6.2\).)34 b(Under)19 b(some)f(restrictions)f(on)i(the)g(quadratic)e(part)i Fz(H)3342 3592 y Fv(0)515 3679 y FA(and)g(on)g(the)h(higher-order)d (part)i Fz(h)p FA(,)i(it)f(is)f(pro)n(v)n(ed)f(that)i(if)g(the)g (initial)f(data)g Fz(u)2884 3691 y Fv(0)2941 3679 y FA(is)g(an)g Fz(")p FA(-small)515 3779 y(and)h(`v)n(ery')g(smo)r(oth)h(function,)i (then)e(a)f(solution)h(sta)n(ys)e(v)n(ery)h(close)g(to)h(the)g(corresp) r(onding)515 3879 y(in)n(v)-5 b(arian)n(t)30 b(torus)g(of)h(the)g (linear)f(system)h(with)h(the)f(hamiltonian)f Fz(H)2712 3891 y Fv(0)2750 3879 y FA(,)i(during)e(the)i(time)515 3978 y(whic)n(h)38 b(is)f(p)r(olynomially)h(large)e(in)i Fz(")1722 3948 y Fu(\000)p Fv(1)1811 3978 y FA(,)j(or)c(ev)n(en)h(exp)r (onen)n(tially)f(large.)66 b(This)38 b(result)515 4078 y(is)33 b(obtained)g(either)g(under)g(the)g(`quasi-Nekhoroshev')e (condition)i(that)g(the)h(sp)r(ectrum)515 4177 y(of)g(the)h(op)r (erator)e Fz(A)i FA(is)g(`highly)f(non-resonan)n(t',)h(or)f(under)g (the)h(opp)r(osite)g(assumption)515 4277 y(\(needed)28 b(to)g(apply)g(the)g(Lo)r(c)n(hak{Niederman)e(tec)n(hnique\))i(that)h (the)f(sp)r(ectrum)g(is)g(`v)n(ery)515 4377 y(resonan)n(t'.)61 b(In)37 b(particular,)g(the)g(follo)n(wing)e(result)h(is)g(pro)n(v)n (ed)f(in)h([Bam99a)o(])g(\(also)f(see)515 4476 y([P\177)-42 b(os99)n(,)19 b(Bou00)n(]\):)33 b(Let)20 b(us)f(consider)f(the)i(NLS)f (equation)g(\(5.24\))f(in)i(the)f(scale)g Fy(f)p Fz(H)3096 4446 y Fw(s)3089 4497 y Fv(0)3130 4476 y FA(\(0)p Fz(;)28 b(\031)s FA(\))p Fy(g)515 4587 y FA(of)37 b(o)r(dd)g(2)p Fz(\031)s FA(-p)r(erio)r(dic)g(functions.)66 b(Assume)37 b(that)h Fz(u)2214 4599 y Fv(0)2251 4587 y FA(\()p Fz(x)p FA(\))i(=)2506 4525 y Fq(P)2593 4545 y Fw(N)2593 4612 y(k)q Fv(=1)2732 4587 y Fz(u)2780 4599 y Fw(k)q Fv(0)2867 4587 y FA(sin)14 b Fz(k)s(x)p FA(,)40 b(denote)515 4686 y Fz(")23 b FA(=)f Fy(j)p Fz(u)735 4698 y Fv(0)772 4686 y FA(\()p Fz(x)p FA(\))p Fy(j)906 4698 y Fw(L)952 4706 y Ft(2)1012 4686 y Fy(\034)h FA(1)g(and)g(write)g(the)h(solution)f Fz(u)p FA(\()p Fz(t;)k(x)p FA(\))e(of)k(\(5.24\))23 b(as)f Fz(u)h FA(=)2849 4624 y Fq(P)2950 4686 y Fz(u)2998 4698 y Fw(k)3038 4686 y FA(\()p Fz(t)p FA(\))14 b(sin)h Fz(k)s(x)p FA(.)p 515 4740 1146 4 v 607 4793 a Fm(8)642 4817 y Fl(Indep)r(enden)n (tly)28 b(this)d(result)g(w)n(as)g(obtained)h(in)f([BF)n(G98)q(])f(b)n (y)i(means)e(of)h(the)h(Nekhoroshev's)g(tec)n(h-)515 4896 y(niques.)1905 5255 y FA(26)p eop PStoPSsaved restore %%Page: (26,27) 14 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 27 26 bop 515 523 a FA(Then)35 b(there)g(exist)g Fz(")1202 535 y Fu(\003)1276 523 y Fz(>)h FA(0)e(and)h(constan)n(ts)g Fz(C)2057 535 y Fv(1)2094 523 y Fz(;)14 b(C)2190 535 y Fv(2)2264 523 y Fz(>)35 b FA(0)g(suc)n(h)g(that)g(for)g Fz(")h(<)f(")3172 535 y Fu(\003)3245 523 y FA(and)515 623 y Fy(j)p Fz(t)p Fy(j)23 b(\024)g Fz(C)761 635 y Fv(1)812 623 y FA(exp\()p Fz(")1010 635 y Fu(\003)1048 623 y Fz(=")p FA(\))1161 593 y Fv(1)p Fw(=)l(N)1310 623 y FA(=:)g Fz(T)1470 635 y Fw(")1532 623 y FA(w)n(e)28 b(ha)n(v)n(e)1324 761 y Fu(1)1297 786 y Fq(X)1297 965 y Fw(k)q Fv(=1)1431 798 y Fq(\000)1469 865 y Fy(j)p Fz(u)1540 877 y Fw(k)1581 865 y FA(\()p Fz(t)p FA(\))p Fy(j)1698 831 y Fv(2)1754 865 y Fy(\000)18 b(j)p Fz(u)1908 877 y Fw(k)q Fv(0)1982 865 y Fy(j)2005 831 y Fv(2)2042 798 y Fq(\001)2080 814 y Fv(2)2140 865 y Fy(\024)23 b Fz(C)2287 877 y Fv(2)2325 865 y Fz(")2364 831 y Fv(4+1)p Fw(=)l(N)2574 865 y Fz(:)611 b FA(\(6.3\))639 1139 y(Let)30 b(us)g(set)g Fz(T)1092 1109 y Fw(N)1181 1139 y FA(=)d Fy(f)p Fz(u)p FA(\()p Fz(x)p FA(\))g(=)1593 1076 y Fq(P)1680 1097 y Fw(N)1680 1164 y(k)q Fv(=1)1819 1139 y Fz(u)1867 1151 y Fw(k)1921 1139 y FA(sin)14 b Fz(k)s(x)27 b Fy(j)g(j)p Fz(u)2278 1151 y Fw(k)2319 1139 y Fy(j)g FA(=)g Fy(j)p Fz(u)2532 1151 y Fw(k)2567 1159 y Ft(0)2603 1139 y Fy(jg)p Fz(:)j FA(This)g(is)g(an)f Fz(n)p FA(-torus)515 1238 y(of)e(diameter)g Fy(\030)c Fz(")k FA(and)h(\(6.3\))f(implies)h(that)1224 1414 y(dist)1358 1426 y Fw(H)1416 1406 y Fn(s)1412 1444 y Ft(0)1453 1414 y FA(\()p Fz(u)p FA(\()p Fz(t)p FA(\))p Fz(;)g(T)1739 1380 y Fw(n)1783 1414 y FA(\))23 b Fy(\024)g Fz(C)1985 1426 y Fw(s)2021 1414 y Fz(")2060 1380 y Fv(1+1)p Fw(=)l(N)2316 1414 y Fy(8)14 b(j)p Fz(t)p Fy(j)22 b(\024)h Fz(T)2612 1426 y Fw(")2647 1414 y Fz(;)515 1590 y FA(if)k Fz(s)22 b(<)h Fy(\000)p FA(1)p Fz(=)p FA(4.)35 b(Th)n(us,)26 b(during)g(the)h(time)f Fz(T)1860 1602 y Fw(")1922 1590 y FA(the)g(tra)5 b(jectory)25 b Fz(u)p FA(\()p Fz(t)p FA(\))h(remains)g(v)n(ery)f(close)g(to)515 1689 y(its)i(pro)5 b(jection)25 b(to)i Fz(T)1185 1659 y Fw(N)1247 1689 y FA(.)37 b(The)26 b(latter)h(is)g(a)f(tra)5 b(jectory)25 b(of)i(an)f Fz(N)9 b FA(-dimensional)26 b(dynamical)515 1789 y(system,)32 b(so)f(the)h(time)g(of)f(its)h(return)f(to)g(a)g Fz(\032")p FA(-neigh)n(b)r(ourho)r(o)r(d)f(\()p Fz(\032)g Fy(\034)f FA(1\))j(of)f(the)h(initial)515 1888 y(p)r(oin)n(t)24 b(`should')g(b)r(e)h(of)f(order)f Fz(\032)1492 1858 y Fu(\000)p Fw(N)1607 1888 y FA(.)35 b(Same)24 b(is)h(true)f(for)f(the)i (tra)5 b(jectory)23 b Fz(u)p FA(\()p Fz(t)p FA(\),)i(if)f Fz(")g FA(is)g(small)515 1988 y(in)e(terms)g(of)h Fz(\032)p FA(.)35 b(The)22 b(phenomenon)g(of)g(the)h(pathologically)e(go)r(o)r(d) g(recurrence)g(prop)r(erties)515 2088 y(of)i(small-amplitude)g(tra)5 b(jectories)21 b(of)i(some)g(non-in)n(tegrable)e(1D)i(HPDEs)g(is)g(w)n (ell)g(kno)n(wn)515 2187 y(from)36 b(n)n(umerics)g(\(e.g.,)j(see)e ([ZIS79)o(]\).)64 b(W)-7 b(e)38 b(ha)n(v)n(e)d(seen)i(that)g(the)g (quasi-Nekhoroshev)515 2287 y(theorems)27 b(as)g(ab)r(o)n(v)n(e)f (explain)h(it)h(up)g(to)f(some)g(extend.)515 2560 y FC(7)134 b(In)l(v)-7 b(arian)l(t)46 b(Gibbs)f(measures)515 2742 y FA(If)32 b(equation)f(\(4.1\))h(is)g(a)f(\014nite-dimensional)h (Hamiltonian)g(system)f(with)i Fz(u)c FA(=)h(\()p Fz(p;)14 b(q)s FA(\))31 b Fy(2)515 2842 y FA(\()p Fx(R)601 2812 y Fv(2)p Fw(n)685 2842 y Fz(;)d(dp)10 b Fy(^)g Fz(dq)s FA(\),)25 b(then)f(an)n(y)f(measure)f Fz(f)9 b FA(\()p Fz(H)e FA(\()p Fz(p;)14 b(q)s FA(\)\))g Fz(dpdq)27 b FA(suc)n(h)c(that)g(the)h(function)g Fz(f)19 b Fy(\016)10 b Fz(H)29 b FA(is)515 2941 y(Leb)r(esgue-in)n(tegrable,)e(is)i(in)n(v) -5 b(arian)n(t)28 b(for)h(the)g(equation.)41 b(The)29 b(most)g(imp)r(ortan)n(t)g(among)515 3041 y(these)h(measures)e(is)i (the)h(Gibbs)f(measure)f Fz(e)1928 3011 y Fu(\000)p Fw(H)2042 3041 y Fz(dpdq)34 b FA(\(the)c(hamiltonian)g Fz(H)37 b FA(is)30 b(assumed)515 3141 y(to)f(gro)n(w)e(to)h(in\014nit)n(y)i (with)f Fy(j)p FA(\()p Fz(p;)14 b(q)s FA(\))p Fy(j)p FA(\).)42 b(No)n(w)28 b(let)h(us)g(consider)f(an)h(HPDE)g(\(4.1\))o(.) 41 b(Sa)n(y)-7 b(,)29 b(the)515 3240 y(zero-mass)c Fz(\036)943 3210 y Fv(4)981 3240 y FA(-equation)1299 3416 y(\177)-47 b Fz(u)23 b FA(=)f Fz(u)1500 3428 y Fw(xx)1598 3416 y Fy(\000)c Fz(u)1729 3382 y Fv(3)1766 3416 y Fz(;)96 b(u)23 b FA(=)g Fz(u)p FA(\()p Fz(t;)14 b(x)p FA(\))p Fz(;)42 b(x)23 b Fy(2)h Fz(S)2540 3382 y Fv(1)2577 3416 y Fz(:)515 3592 y FA(This)j(equation)g(is)h(equiv)-5 b(alen)n(t)27 b(to)h(the)g(system)1538 3762 y(_)-38 b Fz(u)23 b FA(=)f Fy(\000)p Fz(B)t(v)s(;)1535 3897 y FA(_)-35 b Fz(v)26 b FA(=)d Fz(B)t(u)18 b FA(+)g Fz(B)1960 3862 y Fu(\000)p Fv(1)2049 3897 y FA(\()p Fz(u)2129 3862 y Fv(3)2185 3897 y Fy(\000)g Fz(u)p FA(\))c Fz(;)3208 3830 y FA(\(7.1\))515 4084 y(where)28 b Fz(B)h FA(=)938 4017 y Fy(p)p 1007 4017 213 4 v 67 x FA(1)18 b Fy(\000)g FA(\001.)40 b(Denoting)29 b Fz(\030)g FA(=)c(\()p Fz(u;)14 b(v)s FA(\))29 b(w)n(e)f(can)h(see)f (that)h(this)g(is)g(a)f(Hamiltonian)515 4184 y(system)h(in)g(the)h (symplectic)f(scale)g(\()p Fy(f)p Fz(Z)1777 4196 y Fw(s)1837 4184 y FA(=)d Fz(H)2004 4154 y Fw(s)p Fv(+1)p Fw(=)p Fv(2)2190 4184 y FA(\()p Fx(T)2278 4154 y Fv(2)2315 4184 y FA(;)14 b Fx(R)2406 4154 y Fv(2)2449 4184 y FA(\))p Fy(g)p Fz(;)g(\013)2613 4196 y Fv(2)2676 4184 y FA(=)p 2766 4117 55 4 v 25 w Fz(J)8 b(d\030)24 b Fy(^)c Fz(d\030)t FA(\),)31 b(where)515 4283 y Fz(J)8 b FA(\()p Fz(u;)14 b(v)s FA(\))23 b(=)g(\()p Fy(\000)p Fz(v)s(;)14 b(u)p FA(\),)27 b(with)h(the)g(hamiltonian)1008 4501 y Fz(H)7 b FA(\()p Fz(\030)t FA(\))23 b(=)1309 4445 y(1)p 1309 4482 42 4 v 1309 4558 a(2)1360 4501 y Fy(k)p Fz(\030)t Fy(k)1484 4467 y Fv(2)1484 4522 y(0)1539 4501 y FA(+)1622 4388 y Fq(Z)1719 4434 y(\000)1767 4445 y FA(1)p 1767 4482 V 1767 4558 a(4)1819 4501 y Fy(j)p Fz(u)p Fy(j)1913 4467 y Fv(4)1968 4501 y Fy(\000)2061 4445 y FA(1)p 2061 4482 V 2061 4558 a(2)2113 4501 y Fy(j)p Fz(u)p Fy(j)2207 4467 y Fv(2)2243 4434 y Fq(\001)2295 4501 y Fz(dx;)98 b(\030)27 b FA(=)c(\()p Fz(u;)14 b(v)s FA(\))g Fz(:)515 4731 y FA(Here)26 b Fy(k)17 b(\001)h(k)852 4743 y Fv(0)916 4731 y FA(is)27 b(the)g(norm)g(in)g(the)g(space)g Fz(H)1894 4701 y Fv(1)p Fw(=)p Fv(2)1998 4731 y FA(\()p Fz(S)2086 4701 y Fv(1)2123 4731 y FA(;)14 b Fx(R)2214 4701 y Fv(2)2257 4731 y FA(\))28 b(\(cf.)37 b(section)27 b(8.3\).)36 b(The)27 b(natural)515 4831 y(question)g(is)g(if)i(the)f(formal)e(expression) 1699 5006 y Fz(\026)d FA(=)g Fz(e)1899 4972 y Fu(\000)p Fw(H)t Fv(\()p Fw(\030)r Fv(\))2111 5006 y Fz(d\030)1018 b FA(\(7.2\))1905 5255 y(27)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 28 27 bop 515 523 a FA(de\014nes)33 b(a)f(measure)g(in)h(a)f(suitable)h (function)g(space)f(\004)g(=)g Fy(f)p Fz(\030)t FA(\()p Fz(x)p FA(\))p Fy(g)p FA(,)i(in)n(v)-5 b(arian)n(t)32 b(for)g(\015o)n(w-)515 623 y(maps)j(of)g(equation)f(\(7.1\).)60 b(Since)35 b(the)g(Leb)r(esgue)g(measure)f Fz(d\030)40 b FA(do)r(es)35 b(not)g(exist)g(in)g(an)515 722 y (in\014nite-dimensional)26 b(function)h(space,)f(then)h(to)f(mak)n(e)g (the)h(r.h.s.)36 b(of)d(\(7.2\))26 b(meaningful)515 822 y(w)n(e)h(write)g(it)h(as)1333 922 y Fz(\026)23 b FA(=)g Fz(e)1533 887 y Fu(\000)1596 840 y Fd(R)1641 887 y Fv(\()1677 865 y Ft(1)p 1677 874 29 3 v 1677 907 a(4)1715 887 y Fu(j)p Fw(u)p Fu(j)1794 862 y Ft(4)1826 887 y Fu(\000)1888 865 y Ft(1)p 1888 874 V 1888 907 a(2)1926 887 y Fu(j)p Fw(u)p Fu(j)2005 862 y Ft(2)2037 887 y Fv(\))11 b Fw(dx)2151 922 y Fz(e)2190 887 y Fu(\000)2251 865 y Ft(1)p 2251 874 V 2251 907 a(2)2289 887 y Fu(k)p Fw(\030)r Fu(k)2389 862 y Ft(2)2389 904 y(0)2440 922 y Fz(d\030)18 b(:)515 1071 y FA(No)n(w)23 b(exp)14 b Fy(\000)916 1038 y Fv(1)p 915 1052 34 4 v 915 1100 a(2)958 1071 y Fy(k)p Fz(\030)t Fy(k)1082 1041 y Fv(2)1082 1092 y(0)1133 1071 y Fz(d\030)28 b FA(is)c(a)g(w)n(ell-de\014ned)f(Gaussian)g(measure,)h(supp)r(orted)g (b)n(y)g(a)f(suitable)515 1171 y(space)37 b(\004,)j(formed)e(b)n(y)f (functions)h(of)g(lo)n(w)f(smo)r(othness,)i(and)f(0)h Fz(<)g(p)p FA(\()p Fz(\030)t FA(\))i Fy(\024)e Fz(C)6 b FA(,)41 b(where)515 1282 y Fz(p)p FA(\()p Fz(\030)t FA(\))23 b(=)g Fz(e)811 1251 y Fu(\000)874 1205 y Fd(R)919 1251 y Fv(\()954 1229 y Ft(1)p 954 1238 29 3 v 954 1271 a(4)993 1251 y Fu(j)p Fw(u)p Fu(j)1072 1226 y Ft(4)1104 1251 y Fu(\000)1166 1229 y Ft(1)p 1166 1238 V 1166 1271 a(2)1204 1251 y Fu(j)p Fw(u)p Fu(j)1283 1226 y Ft(2)1315 1251 y Fv(\))11 b Fw(dx)1429 1282 y FA(.)37 b(Therefore)26 b(if)639 1381 y(i\))i Fz(p)p FA(\()p Fz(\030)t FA(\))g(is)g(a)f(Borel)g (function)h(on)f(\004,)515 1481 y(then)32 b Fz(\026)g FA(is)g(a)g(w)n(ell-de\014ned)g(Borel)f(measure)g(on)g(\004.)51 b(T)-7 b(o)32 b(c)n(hec)n(k)f(that)h(it)h(is)f(in)n(v)-5 b(arian)n(t)31 b(for)515 1580 y(equation)c(\(7.1\))g(w)n(e)g(ha)n(v)n (e)g(to)g(v)n(erify)g(that)639 1680 y(ii\))k(the)f(\015o)n(w-maps)e(of) 36 b(\(7.1\))30 b(are)f(w)n(ell-de\014ned)g(on)h(supp)14 b Fz(\026)30 b FA(and)f(preserv)n(e)g(the)h(mea-)515 1780 y(sure.)515 1879 y(The)22 b(corresp)r(onding)e(result)i(w)n(as)f (\014rst)h(pro)n(v)n(ed)f(b)n(y)g(F)-7 b(riedlander)22 b([F)-7 b(ri85)o(].)35 b(Unfortunately)-7 b(,)515 1979 y(his)31 b(argumen)n(ts)f(con)n(tain)h(serious)f(\015a)n(ws.)48 b(Complete)31 b(pro)r(ofs)g(app)r(eared)f(later)h(in)h(w)n(orks)515 2079 y(of)37 b(Zhidk)n(o)n(v,)i(McKean{V)-7 b(aninsky)36 b(and)h(Bourgain,)i(see)e(the)h(b)r(o)r(oks)f([Bou99a)n(,)h(Zhi01)o(]) 515 2178 y(and)26 b(references)g(therein.)36 b(Similar)27 b(argumen)n(ts)e(apply)i(to)f(the)h(1D)g(NLS)g(equation)f(\(2.4\),)515 2278 y(where)h(the)h(non-quadratic)e(term)h Fz(q)k FA(satis\014es)c (certain)g(restrictions.)639 2427 y(F)-7 b(or)25 b(higher-dimensional)e (HPDEs)i(the)g(task)f(of)h(constructing)g(the)g(Gibbs)g(measures)515 2527 y(b)r(ecomes)32 b(m)n(uc)n(h)h(more)f(di\016cult.)53 b(The)33 b(only)f(kno)n(wn)g(result)h(is)g(due)g(to)f(Bourgain)f(who) 515 2627 y(pro)n(v)n(ed)26 b(that)i(for)f(the)h(defo)r(cusing)f(2D)h (NLS)g(equation)1463 2809 y Fz(i)14 b FA(_)-37 b Fz(u)22 b FA(=)h(\001)p Fz(u)18 b Fy(\000)g(j)p Fz(u)p Fy(j)1962 2775 y Fv(2)1999 2809 y Fz(u;)96 b(x)24 b Fy(2)f Fx(T)2371 2775 y Fv(2)2408 2809 y Fz(;)515 2992 y FA(the)38 b(Gibbs)g(measure)f (\(7.2\))h(exists)f(and)h(is)f(in)n(v)-5 b(arian)n(t.)67 b(The)38 b(main)g(di\016cult)n(y)g(here)f(is)515 3091 y(the)i(step)g(ii\))g(whic)n(h)f(is)h(no)n(w)f(based)g(on)h(highly)f (non)n(trivial)g(results)g(on)g(regularit)n(y)f(of)515 3191 y(corresp)r(onding)25 b(\015o)n(w-maps)h(in)i(Sob)r(olev)f(spaces) g(of)g(lo)n(w)g(smo)r(othness;)g(see)g(in)h([Bou99a)n(].)515 3466 y FC(8)134 b(The)45 b(non-squeezing)g(phenomenon)716 3615 y(and)g(symplectic)h(capacit)l(y)515 3814 y Fs(8.1)112 b(The)38 b(Gromo)m(v)e(theorem)515 3967 y FA(Let)f(\()p Fx(R)757 3937 y Fv(2)p Fw(n)841 3967 y Fz(;)14 b(\014)925 3979 y Fv(2)962 3967 y FA(\))35 b(b)r(e)h(the)f(space)f Fx(R)1583 3937 y Fv(2)p Fw(n)1702 3967 y FA(=)h Fy(f)p Fz(x)1891 3979 y Fv(1)1928 3967 y Fz(;)14 b(x)2012 3979 y Fu(\000)p Fv(1)2101 3967 y Fz(;)g(:)g(:)g(:)g(;)g(x)2333 3979 y Fu(\000)p Fw(n)2431 3967 y Fy(g)34 b FA(with)h(the)g(Darb)r(oux) g(sym-)515 4066 y(plectic)d(form)g Fz(\014)1030 4078 y Fv(2)1098 4066 y FA(=)1193 4004 y Fq(P)1294 4066 y Fz(dx)1384 4078 y Fw(j)1442 4066 y Fy(^)21 b Fz(dx)1608 4078 y Fu(\000)p Fw(j)1696 4066 y FA(.)50 b(By)32 b Fz(B)1967 4078 y Fw(r)2004 4066 y FA(\()p Fz(x)p FA(\))g(=)e Fz(B)2305 4078 y Fw(r)2341 4066 y FA(\()p Fz(x)p FA(;)14 b Fx(R)2512 4036 y Fv(2)p Fw(n)2596 4066 y FA(\))33 b(and)f Fz(C)2892 4036 y Fw(j)2886 4087 y(\032)2958 4066 y FA(=)e Fz(C)3118 4036 y Fw(j)3112 4087 y(\032)3153 4066 y FA(\()p Fx(R)3239 4036 y Fv(2)p Fw(n)3324 4066 y FA(\),)515 4166 y(1)22 b Fy(\024)h Fz(j)28 b Fy(\024)23 b Fz(n)p FA(,)k(w)n(e)g(denote)h(the)g (follo)n(wing)e(balls)i(and)f(cylinders)g(in)h Fx(R)2656 4136 y Fv(2)p Fw(n)2740 4166 y FA(:)639 4349 y Fz(B)702 4361 y Fw(r)738 4349 y FA(\()p Fz(x)p FA(\))c(=)f Fy(f)p Fz(y)i Fy(j)e(j)p Fz(y)f Fy(\000)c Fz(x)p Fy(j)23 b Fz(<)g(r)r Fy(g)p Fz(;)180 b(C)1814 4314 y Fw(j)1808 4369 y(\032)1872 4349 y FA(=)23 b Fy(f)p Fz(y)i FA(=)e(\()p Fz(y)2229 4361 y Fv(1)2266 4349 y Fz(;)14 b(:)g(:)g(:)g(;)g(y)2492 4361 y Fu(\000)p Fw(n)2588 4349 y FA(\))24 b Fy(j)f Fz(y)2734 4314 y Fv(2)2731 4369 y Fw(j)2789 4349 y FA(+)18 b Fz(y)2916 4314 y Fv(2)2913 4369 y Fu(\000)p Fw(j)3023 4349 y Fz(<)k(\032)3153 4314 y Fv(2)3191 4349 y Fy(g)p Fz(:)639 4531 y FA(The)j(famous)g Fp(\(non-\)sque)l(ezing)i(the)l(or)l(em)32 b FA(b)n(y)24 b(M.)i(Gromo)n(v)d([Gro85)o(])i(states)f(that)i(if)f Fz(f)515 4631 y FA(is)31 b(a)h(symplectomorphism)e Fz(f)18 b FA(:)29 b Fz(B)1604 4643 y Fw(r)1641 4631 y FA(\()p Fz(x)p FA(\))i Fy(!)f Fx(R)1950 4601 y Fv(2)p Fw(n)2066 4631 y FA(suc)n(h)h(that)h(its)g(range)e(b)r(elongs)h(to)h(some)515 4731 y(cylinder)24 b Fz(x)875 4743 y Fv(1)924 4731 y FA(+)12 b Fz(C)1066 4700 y Fw(j)1060 4751 y(\032)1101 4731 y FA(,)25 b Fz(x)1196 4743 y Fv(1)1257 4731 y Fy(2)e Fx(R)1389 4700 y Fv(2)p Fw(n)1474 4731 y FA(,)i(then)g Fz(\032)d Fy(\025)h Fz(r)r FA(.)37 b(F)-7 b(or)23 b(an)i(alternativ)n (e)e(pro)r(of,)h(references)f(and)515 4830 y(discussions)j(see)i([HZ94) o(].)1905 5255 y(28)p eop PStoPSsaved restore %%Page: (28,29) 15 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 29 28 bop 515 523 a Fs(8.2)112 b(In\014nite-dimensional)36 b(case)515 676 y FA(Let)30 b(us)h(consider)e(a)h(symplectic)g(Hilb)r (ert)h(scale)f(\()p Fy(f)p Fz(Z)2207 688 y Fw(s)2242 676 y Fy(g)p Fz(;)14 b(\013)2374 688 y Fv(2)2439 676 y FA(=)p 2531 610 55 4 v 27 w Fz(J)22 b(du)e Fy(^)g Fz(du)p FA(\))31 b(with)g(a)f(basis)515 776 y Fy(f)p Fz(')611 788 y Fw(j)669 776 y Fy(j)23 b Fz(j)28 b Fy(2)23 b Fx(Z)916 788 y Fv(0)948 776 y Fy(g)p FA(.)36 b(W)-7 b(e)26 b(assume)g(that)h (this)f(is)g(a)g(shifted)h(Darb)r(oux)f(scale)f(\(cf.)37 b(Example)26 b(2.4)515 876 y(in)g(section)f(2.2\).)36 b(It)26 b(means)f(that)h(the)g(basis)f(can)g(b)r(e)h(renormalised)e(to) h(a)h(basis)f Fy(f)11 b Fq(e)-57 b Fz(')3158 888 y Fw(j)3216 876 y Fy(j)23 b Fz(j)28 b Fy(2)515 975 y Fx(Z)576 987 y Fv(0)607 975 y Fy(g)j FA(\(eac)n(h)42 b Fq(e)-57 b Fz(')957 987 y Fw(j)1024 975 y FA(is)32 b(prop)r(ortional)d(to)j Fz(')1753 987 y Fw(j)1788 975 y FA(\))g(whic)n(h)f(is)h(a)f(Darb)r(oux) g(basis)g(for)g(the)g(form)h Fz(\013)3342 987 y Fv(2)515 1075 y FA(and)27 b(a)g(Hilb)r(ert)i(basis)d(of)i(some)f(space)g Fz(Z)1817 1087 y Fw(d)1855 1075 y FA(:)1015 1257 y Fy(h)11 b Fq(e)-57 b Fz(')1101 1269 y Fw(j)1136 1257 y Fz(;)25 b Fq(e)-57 b Fz(')1227 1269 y Fw(k)1268 1257 y Fy(i)1300 1269 y Fw(d)1362 1257 y FA(=)23 b Fz(\016)1487 1269 y Fw(j;k)1575 1257 y Fz(;)97 b(\013)1748 1269 y Fv(2)1785 1257 y FA([)11 b Fq(e)-57 b Fz(')1862 1269 y Fw(j)1897 1257 y Fz(;)25 b Fq(e)-57 b Fz(')1988 1269 y Fu(\000)p Fw(k)2081 1257 y FA(])23 b(=)g(sgn)13 b Fz(j)19 b(\016)2439 1269 y Fw(j;k)2693 1257 y Fy(8)p Fz(j;)14 b(k)s(:)328 b FA(\(8.1\))515 1440 y(These)27 b(relations)f(imply)i(that)1138 1623 y Fz(\013)1191 1635 y Fv(2)1228 1623 y FA([)p Fz(\030)t(;)14 b(\021)s FA(])23 b(=)g Fy(h)p 1538 1556 V Fz(J)8 b(\030)t(;)14 b(\021)s Fy(i)1745 1635 y Fw(d)1785 1623 y Fz(;)p 1987 1556 V 179 w(J)19 b Fq(e)-57 b Fz(')2095 1635 y Fw(j)2154 1623 y FA(=)22 b(sgn)14 b Fz(j)29 b Fq(e)-57 b Fz(')2482 1635 y Fu(\000)p Fw(j)2653 1623 y Fy(8)p Fz(j:)451 b FA(\(8.2\))515 1805 y(In)28 b(particular,)p 1025 1739 V 26 w Fz(J)j FA(=)23 b Fz(J)8 b FA(.)639 1905 y(Belo)n(w)27 b(w)n(e)g(skip)h(the)g(tildes)f(and)h(re-denote)e(the)i(new)g(basis)f (bac)n(k)g(to)g Fy(f)p Fz(')2980 1917 y Fw(j)3015 1905 y Fy(g)p FA(.)639 2005 y(In)f(this)f(scale)f(w)n(e)h(consider)f(a)h (semilinear)f(Hamiltonian)h(equation)f(with)i(the)f(Hamil-)515 2104 y(tonian)i Fz(H)7 b FA(\()p Fz(u)p FA(\))23 b(=)1082 2072 y Fv(1)p 1082 2086 34 4 v 1082 2133 a(2)1125 2104 y Fy(h)p Fz(Au;)14 b(u)p Fy(i)1384 2116 y Fw(d)1441 2104 y FA(+)k Fz(h)p FA(\()p Fz(u;)c(t)p FA(\).)36 b(Due)28 b(to)g(\(8.2\))f(it)h(can)f(b)r(e)h(written)g(as)1544 2299 y(_)-38 b Fz(u)23 b FA(=)f Fz(J)8 b(Au)19 b FA(+)f Fz(J)8 b Fy(r)2076 2265 y Fw(d)2115 2299 y Fz(h)p FA(\()p Fz(u;)14 b(t)p FA(\))p Fz(;)843 b FA(\(8.3\))515 2482 y(where)27 b Fy(r)824 2451 y Fw(d)890 2482 y FA(signi\014es)g(the)h (gradien)n(t)f(in)h Fz(u)f FA(with)h(resp)r(ect)f(to)h(the)g(scalar)e (pro)r(duct)h(of)h Fz(Z)3258 2494 y Fw(d)3296 2482 y FA(.)639 2581 y(If)41 b(a)g(Hamiltonian)f(PDE)h(is)f(written)h(in)g (the)g(form)g(\(8.3\))o(,)k(then)c(the)g(symplectic)515 2681 y(space)28 b(\()p Fz(Z)827 2693 y Fw(d)865 2681 y Fz(;)14 b(\013)955 2693 y Fv(2)993 2681 y FA(\))29 b(is)f(called)g(the)h Fp(\(Hilb)l(ert\))i(Darb)l(oux)f(phase)i(sp)l(ac) l(e)k FA(for)28 b(this)h(PDE.)f(Belo)n(w)515 2780 y(w)n(e)f(study)h (prop)r(erties)e(of)i(\015o)n(w-maps)e(of)i(equation)f(\(8.3\))g(in)h (its)g(Darb)r(oux)f(phase)g(space.)639 2880 y(Let)h(us)g(assume)e(that) i(the)g(op)r(erator)e Fz(A)i FA(has)f(the)h(form)512 3063 y(\(H1\))42 b Fz(Au)23 b FA(=)943 3000 y Fq(P)1030 3021 y Fu(1)1030 3088 y Fw(j)s Fv(=1)1163 3063 y Fz(\025)1211 3075 y Fw(j)1247 3063 y FA(\()p Fz(u)1327 3075 y Fw(j)1361 3063 y Fz(')1415 3075 y Fw(j)1469 3063 y FA(+)18 b Fz(u)1600 3075 y Fu(\000)p Fw(j)1687 3063 y Fz(')1741 3075 y Fu(\000)p Fw(j)1828 3063 y FA(\))97 b Fy(8)p Fz(u)22 b FA(=)2161 3000 y Fq(P)2263 3063 y Fz(u)2311 3075 y Fw(j)2345 3063 y Fz(')2399 3075 y Fw(j)2448 3063 y FA(,)722 3195 y(where)27 b Fz(\025)1010 3207 y Fw(j)1046 3195 y FA('s)g(are)g(some)g(real)g(n)n (um)n(b)r(ers.)515 3378 y(Then)j Fz(J)8 b(Au)26 b FA(=)1015 3316 y Fq(P)1103 3336 y Fu(1)1103 3403 y Fw(j)s Fv(=1)1236 3378 y Fz(\025)1284 3390 y Fw(j)1319 3378 y FA(\()p Fz(u)1399 3390 y Fu(\000)p Fw(j)1486 3378 y Fz(')1540 3390 y Fu(\000)p Fw(j)1647 3378 y Fy(\000)19 b Fz(u)1779 3390 y Fw(j)1814 3378 y Fz(')1868 3390 y Fw(j)1903 3378 y FA(\),)31 b(so)e(the)h(linear) f(op)r(erators)f Fz(e)2880 3348 y Fw(tJ)5 b(A)3030 3378 y FA(are)29 b(direct)515 3478 y(sums)e(of)h(rotations)e(in)i(the)g (planes)f Fx(R)p Fz(')1771 3490 y Fw(j)1831 3478 y FA(+)18 b Fx(R)p Fz(')2022 3490 y Fu(\000)p Fw(j)2138 3478 y Fy(\032)23 b Fz(Z)2283 3490 y Fw(d)2321 3478 y FA(,)28 b Fz(j)g FA(=)22 b(1)p Fz(;)14 b FA(2)p Fz(;)g(:)g(:)g(:)f FA(.)639 3577 y(W)-7 b(e)28 b(also)f(assume)g(that)h(the)g(gradien)n(t) e(map)h Fy(r)2137 3547 y Fw(d)2176 3577 y Fz(h)h FA(is)f(smo)r(othing:) 512 3760 y(\(H2\))42 b(there)23 b(exists)f Fz(\015)28 b(>)22 b FA(0)h(suc)n(h)f(that)h(ord)13 b Fy(r)1937 3730 y Fw(d)1976 3760 y Fz(h)23 b FA(=)f Fy(\000)p Fz(\015)27 b FA(for)c Fz(s)g Fy(2)g FA([)p Fz(d)9 b Fy(\000)g Fz(\015)c(;)14 b(d)9 b FA(+)g Fz(\015)c FA(].)33 b(Moreo)n(v)n(er,)722 3860 y(the)28 b(maps)1244 3959 y Fy(r)1313 3925 y Fw(d)1352 3959 y Fz(h)9 b FA(:)27 b Fz(Z)1516 3971 y Fw(s)1570 3959 y Fy(\002)18 b Fx(R)29 b Fy(!)23 b Fz(Z)1899 3971 y Fw(s)p Fv(+)p Fw(\015)2024 3959 y Fz(;)180 b(s)23 b Fy(2)g FA([)p Fz(d)c Fy(\000)f Fz(\015)5 b(;)14 b(d)k FA(+)g Fz(\015)5 b FA(])p Fz(;)722 4109 y FA(are)27 b Fz(C)926 4079 y Fv(1)963 4109 y FA(-smo)r(oth)g(and)h(b)r(ounded.)1817 4079 y Fv(9)639 4291 y FA(F)-7 b(or)21 b(an)n(y)f Fz(t)h FA(and)g Fz(T)32 b FA(w)n(e)21 b(denote)g(b)n(y)g Fz(O)1770 4261 y Fw(T)1768 4312 y(t)1844 4291 y FA(an)n(y)f(op)r(en)h(subset)g (of)g(the)h(domain)f(of)g(de\014nition)515 4391 y(of)30 b(the)h(\015o)n(w-map)e Fz(S)1174 4361 y Fw(T)1169 4412 y(t)1257 4391 y FA(in)h Fz(Z)1413 4403 y Fw(d)1452 4391 y FA(,)h(suc)n(h)f(that)h(for)e(eac)n(h)h(b)r(ounded)h(subset)f Fz(Q)e Fy(\032)f Fz(O)3048 4361 y Fw(T)3046 4412 y(t)3131 4391 y FA(the)k(set)515 4428 y Fq(S)584 4516 y Fw(\034)7 b Fu(2)p Fv([)p Fw(t;T)i Fv(])815 4491 y Fz(S)871 4461 y Fw(\034)866 4511 y(t)912 4491 y FA(\()p Fz(Q)p FA(\))28 b(is)f(b)r(ounded)h(in)g Fz(Z)1646 4503 y Fw(d)1684 4491 y FA(.)1707 4461 y Fv(10)639 4607 y FA(In)i(the)f(theorem)g(b)r(elo)n (w)g(the)h(balls)f Fz(B)1855 4619 y Fw(r)1921 4607 y FA(and)g(the)h(cylinders)f Fz(C)2645 4577 y Fw(j)2639 4627 y(\032)2680 4607 y Fz(;)f(j)i Fy(\025)c FA(1,)j(are)g(de\014ned) 515 4707 y(in)f(the)g(same)f(w)n(a)n(y)f(as)h(in)h(section)f(8.1.)p 515 4760 1146 4 v 607 4813 a Fm(9)642 4837 y Fl(i.e.,)22 b(they)j(send)f(b)r(ounded)h(sets)f(to)g(b)r(ounded.)577 4893 y Fm(10)642 4916 y Fl(this)f(set)i(should)e(b)r(e)h(treated)i(as)d (a)h(`regular)f(part)h(of)f(the)i(domain)e(of)g(de\014nition'.)1905 5255 y FA(29)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 30 29 bop 515 523 a FB(Theorem)37 b(8.1.)44 b Fp(L)l(et)34 b(us)h(assume)g(that)h(the)f(assumptions)42 b FA(\(H1\))36 b Fp(and)44 b FA(\(H2\))36 b Fp(hold)g(and)515 623 y(that)26 b(a)h(b)l(al)t(l)h Fz(B)963 635 y Fw(r)1023 623 y FA(=)22 b Fz(B)1173 635 y Fw(r)1210 623 y FA(\()p Fz(u)1290 635 y Fv(0)1327 623 y FA(;)14 b Fz(Z)1421 635 y Fw(d)1460 623 y FA(\))23 b(:=)g Fy(fk)p Fz(y)13 b Fy(\000)f Fz(u)1889 635 y Fv(0)1925 623 y Fy(k)23 b Fz(<)f(r)r Fy(g)27 b Fp(b)l(elongs)g(to)g Fz(O)2628 593 y Fw(T)2626 643 y(t)2707 623 y Fp(to)l(gether)g(with)h(some)515 722 y Fz(")p Fp(-neighb)l(ourho) l(o)l(d,)j Fz(")23 b(>)g FA(0)p Fp(.)38 b(Then)30 b(the)g(r)l(elation) 1536 905 y Fz(S)1592 871 y Fw(T)1587 925 y(t)1644 905 y FA(\()p Fz(B)1739 917 y Fw(r)1776 905 y FA(\))23 b Fy(\032)g Fz(v)1959 917 y Fv(0)2015 905 y FA(+)18 b Fz(C)2163 871 y Fw(j)2157 925 y(\032)2198 905 y FA(\()p Fz(Z)2287 917 y Fw(d)2326 905 y FA(\))850 b(\(8.4\))515 1088 y Fp(with)30 b(some)g Fz(v)947 1100 y Fv(0)1008 1088 y Fy(2)23 b Fz(Z)1143 1100 y Fw(d)1211 1088 y Fp(and)31 b Fz(j)d Fy(\025)22 b FA(1)29 b Fp(implies)j(that)d Fz(\032)23 b Fy(\025)g Fz(r)r Fp(.)515 1254 y(Pr)l(o)l(of.)43 b FA(Without)28 b(lost)g(of)f(generalit)n(y)f(w)n(e)h(ma)n(y)g(assume)g (that)1623 1436 y Fz(v)1663 1448 y Fv(0)1724 1436 y FA(=)c(0)p Fz(;)179 b(j)28 b FA(=)23 b(1)p Fz(:)515 1619 y FA(Arguing)g(b)n(y)g (con)n(tradiction)g(w)n(e)g(assume)g(that)h(\(8.4\))g(holds)f(with)h Fz(\032)f(<)g(r)k FA(and)c(c)n(ho)r(ose)g(an)n(y)515 1719 y Fz(\032)558 1731 y Fv(1)618 1719 y Fy(2)g FA(\()p Fz(\032;)14 b(r)r FA(\).)639 1818 y(F)-7 b(or)34 b Fz(n)h Fy(\025)f FA(1)h(w)n(e)f(denote)h(b)n(y)f Fz(E)1648 1788 y Fv(2)p Fw(n)1761 1818 y FA(the)h(subspace)f(of)h Fz(Z)2424 1830 y Fw(d)2462 1818 y FA(,)i(spanned)d(b)n(y)h(the)g(v)n(ectors)515 1918 y Fy(f)p Fz(')611 1930 y Fw(j)646 1918 y Fz(;)14 b Fy(j)p Fz(j)5 b Fy(j)31 b(\024)h Fz(n)p Fy(g)p FA(,)i(and)e(pro)n (vide)g(it)h(with)h(the)f(usual)g(Darb)r(oux)f(symplectic)h(structure)f (\(it)515 2017 y(is)c(giv)n(en)g(b)n(y)g(the)h(form)g Fz(\013)1328 2029 y Fv(2)1365 2017 y Fy(j)1388 2032 y Fw(E)1440 2016 y Ft(2)p Fn(n)1513 2017 y FA(\).)40 b(By)29 b(\005)1802 2029 y Fw(n)1876 2017 y FA(w)n(e)f(denote)h(the)g (orthogonal)d(pro)5 b(jection)28 b(\005)3286 2029 y Fw(n)3356 2017 y FA(:)515 2117 y Fz(Z)572 2129 y Fw(d)633 2117 y Fy(!)23 b Fz(E)805 2087 y Fv(2)p Fw(n)884 2117 y FA(.)37 b(W)-7 b(e)28 b(set)1406 2217 y Fz(H)1482 2182 y Fw(n)1550 2217 y FA(=)1647 2184 y Fv(1)p 1647 2198 34 4 v 1647 2245 a(2)1690 2217 y Fy(h)p Fz(Au;)14 b(u)p Fy(i)1949 2229 y Fw(d)2006 2217 y FA(+)k Fz(h)p FA(\(\005)2231 2229 y Fw(n)2277 2217 y FA(\()p Fz(u)p FA(\))p Fz(;)c(t)p FA(\))515 2366 y(and)30 b(denote)g(b)n(y)g Fz(S)1123 2336 y Fw(T)1118 2393 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1271 2366 y FA(\015o)n(w-maps)f(of)h(the)h(Hamiltonian)f(v)n(ector)f (\014led)i Fz(V)2870 2378 y Fw(H)2928 2362 y Fn(n)2974 2366 y FA(.)45 b(An)n(y)30 b(map)515 2486 y Fz(S)571 2455 y Fw(T)566 2512 y Fv(\()p Fw(n)p Fv(\))p Fw(t)722 2486 y FA(decomp)r(oses)i(to)i(the)g(direct)f(sum)h(of)g(a)f (symplectomorphism)g(of)g Fz(E)2957 2455 y Fv(2)p Fw(n)3069 2486 y FA(and)h(of)f(a)515 2603 y(linear)i(symplectomorphism)f(of)i Fz(Z)1672 2615 y Fw(d)1734 2603 y Fy(\011)23 b Fz(E)1888 2573 y Fv(2)p Fw(n)1967 2603 y FA(.)60 b(So)36 b(the)g(theorem's)f (assertion)f(with)i(the)515 2702 y(map)h Fz(S)765 2672 y Fw(T)760 2723 y(t)854 2702 y FA(replaced)f(b)n(y)h Fz(S)1372 2672 y Fw(T)1367 2729 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1526 2702 y FA(follo)n(ws)f(from)h(the)g(Gromo)n(v)f(theorem,)j (applied)e(to)g(the)515 2802 y(symplectomorphism)1201 2985 y Fz(E)1267 2950 y Fv(2)p Fw(n)1368 2985 y Fy(!)23 b Fz(E)1540 2950 y Fv(2)p Fw(n)1619 2985 y Fz(;)179 b(x)24 b Fy(7!)f FA(\005)2060 2997 y Fw(n)2106 2985 y Fz(S)2162 2950 y Fw(T)2157 3007 y Fv(\()p Fw(n)p Fv(\))p Fw(t)2279 2985 y FA(\()p Fz(i)p FA(\()p Fz(x)p FA(\))c(+)f Fz(u)2601 2997 y Fv(0)2638 2985 y FA(\))p Fz(;)515 3181 y FA(where)27 b Fz(i)g FA(stands)g(for)g(the)h(em)n(b)r(edding)g(of)g Fz(E)1921 3150 y Fv(2)p Fw(n)2027 3181 y FA(to)f Fz(Z)2185 3193 y Fw(d)2224 3181 y FA(.)639 3280 y(Pro)r(ofs)f(of)i(the)g(t)n(w)n (o)f(easy)f(lemmas)i(b)r(elo)n(w)f(can)g(b)r(e)h(found)g(in)g([Kuk95)n (].)515 3430 y FB(Lemma)19 b(8.2.)33 b Fp(Under)22 b(the)g(the)l(or)l (em's)g(assumptions)g(the)g(maps)h Fz(S)2624 3400 y Fw(T)2619 3456 y Fv(\()p Fw(n)p Fv(\))p Fw(t)2763 3430 y Fp(ar)l(e)f(de\014ne)l (d)g(on)g Fz(B)3342 3442 y Fw(r)515 3542 y Fp(for)28 b Fz(n)23 b Fy(\025)g Fz(n)856 3512 y Fu(0)907 3542 y Fp(with)28 b(some)h(su\016ciently)f(lar)l(ge)h Fz(n)1962 3512 y Fu(0)1985 3542 y Fp(,)f(and)h(ther)l(e)f(exists)f(a)i(se)l (quenc)l(e)e Fz(")3073 3554 y Fw(n)3161 3542 y Fy(\000)-14 b(!)3141 3591 y Fw(n)p Fu(!1)3337 3542 y FA(0)515 3666 y Fp(such)30 b(that)1505 3765 y Fy(k)p Fz(S)1603 3731 y Fw(T)1598 3786 y(t)1654 3765 y FA(\()p Fz(u)p FA(\))19 b Fy(\000)f Fz(S)1924 3731 y Fw(T)1919 3788 y Fv(\()p Fw(n)p Fv(\))p Fw(t)2041 3765 y FA(\()p Fz(u)p FA(\))p Fy(k)23 b(\024)f Fz(")2344 3777 y Fw(n)3208 3765 y FA(\(8.5\))515 3915 y Fp(for)30 b Fz(n)23 b Fy(\025)g Fz(n)858 3885 y Fu(0)911 3915 y Fp(and)30 b(for)h(every)f Fz(u)23 b Fy(2)g Fz(B)1636 3927 y Fw(r)1673 3915 y Fp(.)515 4064 y FB(Lemma)29 b(8.3.)39 b Fp(F)-6 b(or)30 b(any)f Fz(u)23 b Fy(2)g Fz(B)1567 4076 y Fw(r)1633 4064 y Fp(we)30 b(have)59 b Fz(S)2031 4034 y Fw(T)2026 4085 y(t)2083 4064 y FA(\()p Fz(u)p FA(\))23 b(=)g Fz(e)2345 4034 y Fv(\()p Fw(T)9 b Fu(\000)p Fw(t)p Fv(\))p Fw(J)c(A)2618 4064 y Fz(u)17 b FA(+)2776 4043 y Fq(e)2765 4064 y Fz(S)2821 4034 y Fw(T)2816 4085 y(t)2873 4064 y FA(\()p Fz(u)p FA(\))p Fz(;)29 b Fp(wher)l(e)3283 4043 y Fq(e)3271 4064 y Fz(S)3327 4034 y Fw(T)3322 4085 y(t)515 4174 y Fp(is)h(a)g Fz(C)741 4144 y Fv(1)779 4174 y Fp(-smo)l(oth)f(map)i(in)f(the)g(sc)l(ale)g Fy(f)p Fz(Z)1811 4186 y Fw(s)1846 4174 y Fy(g)f Fp(and)h FA(ord)2224 4153 y Fq(e)2212 4174 y Fz(S)2268 4144 y Fw(T)2263 4195 y(t)2343 4174 y FA(=)23 b Fy(\000)p Fz(\015)34 b Fp(for)c Fz(s)23 b Fy(2)h FA([)p Fz(d)18 b Fy(\000)g Fz(\015)5 b(;)14 b(d)19 b FA(+)f Fz(\015)5 b FA(])p Fp(.)639 4374 y FA(No)n(w)38 b(w)n(e)g(con)n(tin)n(ue)f(the)i(pro)r(of)e(of)h (the)h(theorem.)68 b(Since)38 b(its)g(assertion)f(holds)h(for)515 4473 y(an)n(y)29 b(map)g Fz(S)916 4443 y Fw(T)911 4500 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1063 4473 y FA(\()p Fz(n)e Fy(\025)f Fz(n)1313 4443 y Fu(0)1336 4473 y FA(\))k(and)f(since)h(the)g (ball)f Fz(B)2138 4485 y Fw(r)2205 4473 y FA(b)r(elongs)g(to)h(this)f (map's)h(domain)f(of)515 4585 y(de\014nition)i(\(see)f(Lemma)h(8.2\),)g (then)g(for)f(eac)n(h)g Fz(n)e Fy(\025)f Fz(n)2281 4555 y Fu(0)2335 4585 y FA(there)k(exists)f(a)g(p)r(oin)n(t)h Fz(u)3123 4597 y Fw(n)3196 4585 y Fy(2)d Fz(B)3342 4597 y Fw(r)515 4685 y FA(suc)n(h)f(that)h Fz(S)938 4655 y Fw(T)933 4711 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1055 4685 y FA(\()p Fz(u)1135 4697 y Fw(n)1180 4685 y FA(\))33 b Fz(=)-52 b Fy(2)24 b Fz(C)1379 4655 y Fv(1)1373 4705 y Fw(\032)1407 4713 y Ft(1)1444 4685 y FA(\(0\).)37 b(That)28 b(is,)1602 4891 y Fy(j)p FA(\005)1687 4903 y Fv(1)1724 4891 y Fz(S)1780 4857 y Fw(T)1775 4914 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1898 4891 y FA(\()p Fz(u)1978 4903 y Fw(n)2023 4891 y FA(\))p Fy(j)23 b(\025)g Fz(\032)2232 4903 y Fv(1)2269 4891 y Fz(:)916 b FA(\(8.6\))1905 5255 y(30)p eop PStoPSsaved restore %%Page: (30,31) 16 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 31 30 bop 515 523 a FA(By)35 b(the)g(w)n(eak)f(compactness)g(of)h(a)g (Hilb)r(ert)g(ball,)i(w)n(e)e(can)g(\014nd)g(a)g(w)n(eakly)f(con)n(v)n (erging)515 623 y(subsequence)1685 722 y Fz(u)1733 734 y Fw(n)1774 742 y Fn(j)1831 722 y Fz(*)23 b(u)g Fy(2)g Fz(B)2149 734 y Fw(r)2186 722 y Fz(;)999 b FA(\(8.7\))515 871 y(so)1442 970 y Fz(u)1490 982 y Fw(n)1531 990 y Fn(j)1589 970 y Fy(!)23 b Fz(u)82 b FA(strongly)26 b(in)i Fz(Z)2300 982 y Fw(d)p Fu(\000)p Fw(\015)2429 970 y Fz(:)515 1137 y FA(Due)k(to)g(Lemma)g(8.3)f(this)h(implies)g(that)1886 1116 y Fq(e)1875 1137 y Fz(S)1931 1107 y Fw(T)1926 1157 y(t)1983 1137 y FA(\()p Fz(u)2063 1149 y Fw(n)2104 1157 y Fn(j)2139 1137 y FA(\))e Fy(!)2326 1116 y Fq(e)2315 1137 y Fz(S)2371 1107 y Fw(T)2366 1157 y(t)2423 1137 y FA(\()p Fz(u)p FA(\))i(in)g Fz(Z)2725 1149 y Fw(d)2764 1137 y FA(,)h(and)f(using)g(\(8.7\))515 1236 y(w)n(e)27 b(obtain)g(the)h(con)n(v)n(ergence:)1613 1417 y Fz(S)1669 1383 y Fw(T)1664 1438 y(t)1721 1417 y FA(\()p Fz(u)1801 1429 y Fw(n)1842 1437 y Fn(j)1877 1417 y FA(\))23 b Fz(*)g(S)2094 1383 y Fw(T)2089 1438 y(t)2146 1417 y FA(\()p Fz(u)p FA(\))p Fz(:)927 b FA(\(8.8\))639 1598 y(Noting)29 b(that)f Fy(j)p FA(\005)1180 1610 y Fv(1)1218 1598 y Fz(S)1274 1568 y Fw(T)1269 1618 y(t)1326 1598 y FA(\()p Fz(u)1406 1610 y Fw(n)1451 1598 y FA(\))p Fy(j)d FA(=)f Fy(j)p FA(\005)1705 1610 y Fv(1)1743 1598 y Fz(S)1799 1568 y Fw(T)1794 1624 y Fv(\()p Fw(n)p Fv(\))p Fw(t)1916 1598 y Fz(u)1964 1610 y Fw(n)2028 1598 y FA(+)18 b(\005)2173 1610 y Fv(1)2211 1598 y FA(\()p Fz(S)2299 1568 y Fw(T)2294 1618 y(t)2370 1598 y Fy(\000)h Fz(S)2510 1568 y Fw(T)2505 1624 y Fv(\()p Fw(n)p Fv(\))p Fw(t)2627 1598 y FA(\))p Fz(u)2707 1610 y Fw(n)2752 1598 y Fy(j)29 b FA(and)f(using)h(\(8.6\),) 515 1710 y(\(8.5\))e(w)n(e)g(get:)1323 1809 y Fy(j)p FA(\005)1408 1821 y Fv(1)1446 1809 y Fz(S)1502 1775 y Fw(T)1497 1830 y(t)1554 1809 y FA(\()p Fz(u)1634 1821 y Fw(n)1679 1809 y FA(\))p Fy(j)c(\025)g Fz(\032)1888 1821 y Fv(1)1944 1809 y Fy(\000)18 b Fz(")2066 1821 y Fw(n)2111 1809 y Fz(;)180 b(n)22 b Fy(\025)h Fz(n)2524 1775 y Fu(0)2547 1809 y Fz(:)638 b FA(\(8.9\))639 1957 y(Since)39 b(b)n(y)g(\(8.8\))e(\005)1264 1969 y Fv(1)1302 1957 y Fz(S)1358 1927 y Fw(T)1353 1978 y(t)1410 1957 y FA(\()p Fz(u)1490 1969 y Fw(n)1531 1977 y Fn(j)1566 1957 y FA(\))k Fy(!)f FA(\005)1824 1969 y Fv(1)1862 1957 y Fz(S)1918 1927 y Fw(T)1913 1978 y(t)1970 1957 y FA(\()p Fz(u)p FA(\))e(in)h Fz(E)2294 1927 y Fv(2)2331 1957 y FA(,)i(then)d(due)h(to)f(\(8.9\))g(w)n(e)f(ha)n(v)n(e)515 2057 y Fy(j)p FA(\005)600 2069 y Fv(1)637 2057 y Fz(S)693 2027 y Fw(T)688 2078 y(t)745 2057 y FA(\()p Fz(u)p FA(\))p Fy(j)29 b(\025)f Fz(\032)1045 2069 y Fv(1)1083 2057 y FA(.)47 b(This)31 b(con)n(tradicts)e(\(8.4\))i(b)r(ecause)g Fz(\032)2332 2069 y Fv(1)2398 2057 y Fz(>)d(\032)p FA(.)47 b(The)31 b(obtained)g(con)n(tra-)515 2157 y(diction)c(pro)n(v)n(es)f (the)i(theorem.)p 3318 2157 4 57 v 3322 2104 50 4 v 3322 2157 V 3372 2157 4 57 v 515 2389 a Fs(8.3)112 b(Examples)515 2542 y Fp(Example)37 b FA(8.4)p Fp(.)k FA(Let)27 b(us)h(consider)f(the) h(nonlinear)e(w)n(a)n(v)n(e)g(equation)1596 2723 y(\177)-47 b Fz(u)22 b FA(=)h(\001)p Fz(u)18 b Fy(\000)1985 2701 y FA(~)1967 2723 y Fz(f)9 b FA(\()p Fz(u)p FA(;)14 b Fz(t;)g(x)p FA(\))p Fz(;)864 b FA(\(8.10\))515 2903 y(where)29 b Fz(u)e FA(=)g Fz(u)p FA(\()p Fz(t;)14 b(x)p FA(\),)32 b Fz(x)27 b Fy(2)h Fx(T)1418 2873 y Fw(n)1463 2903 y FA(.)44 b(The)31 b(function)2049 2881 y(~)2031 2903 y Fz(f)39 b FA(is)30 b(a)g(p)r(olynomial)g(in)g Fz(u)g FA(of)g(a)g(degree)f Fz(D)515 3003 y FA(suc)n(h)j(that)h(its)f(co)r (e\016cien)n(ts)g(are)g(smo)r(oth)g(functions)h(of)f Fz(t)h FA(and)f Fz(x)p FA(.)52 b(W)-7 b(e)33 b(set)f Fz(f)40 b FA(=)3169 2981 y(~)3151 3003 y Fz(f)30 b Fy(\000)21 b Fz(u)p FA(,)515 3102 y(denote)27 b(b)n(y)f Fz(B)31 b FA(the)c(linear)f(op)r(erator)g Fz(B)h FA(=)1874 3036 y Fy(p)p 1944 3036 213 4 v 1944 3102 a FA(1)18 b Fy(\000)g FA(\001)27 b(and)f(write)h(\(8.10\))f(as)g(the)i(system)e(of)515 3202 y(t)n(w)n(o)h(equations:)1535 3296 y(_)-38 b Fz(u)23 b FA(=)g Fy(\000)p Fz(B)t(v)s(;)1537 3431 y FA(_)-35 b Fz(v)26 b FA(=)d Fz(B)t(u)18 b FA(+)g Fz(B)1962 3397 y Fu(\000)p Fv(1)2051 3431 y Fz(f)9 b FA(\()p Fz(u)p FA(;)14 b Fz(t;)g(x)p FA(\))p Fz(:)3167 3364 y FA(\(8.11\))639 3592 y(Let)30 b(us)f(tak)n(e)g(for)g Fy(f)p Fz(Z)1308 3604 y Fw(s)1343 3592 y Fy(g)g FA(the)h(shifted)g(Sob)r(olev)f(scale)g Fz(Z)2400 3604 y Fw(s)2461 3592 y FA(=)d Fz(H)2628 3561 y Fw(s)p Fv(+1)p Fw(=)p Fv(2)2814 3592 y FA(\()p Fx(T)2902 3561 y Fw(n)2947 3592 y FA(;)14 b Fx(R)3038 3561 y Fv(2)3081 3592 y FA(\),)31 b(where)515 3691 y Fy(h)p Fz(\030)t(;)14 b(\021)s Fy(i)700 3703 y Fw(s)778 3691 y FA(=)884 3624 y Fq(R)923 3721 y Fo(T)968 3704 y Fn(n)1021 3691 y Fz(B)1088 3661 y Fv(2)p Fw(s)p Fv(+1)1240 3691 y Fz(\030)30 b Fy(\001)c Fz(\021)17 b(dx)40 b FA(\(its)f(basic)f(scalar)f(pro)r(duct)i(is)g(the) g(scalar)e(pro)r(duct)i(in)515 3803 y Fz(H)591 3773 y Fv(1)p Fw(=)p Fv(2)695 3803 y FA(\).)56 b(W)-7 b(e)34 b(set)g Fz(\013)1144 3815 y Fv(2)1214 3803 y FA(=)p 1312 3736 55 4 v 33 w Fz(J)22 b(d\030)27 b Fy(^)c Fz(d\030)t FA(,)36 b(where)d Fz(J)8 b(\030)38 b FA(=)33 b(\()p Fy(\000)p Fz(v)s(;)14 b(u)p FA(\))34 b(for)f Fz(\030)k FA(=)c(\()p Fz(u;)14 b(v)s FA(\).)56 b(Cho)r(osing)515 3902 y(for)40 b Fy(f)p Fz( )751 3914 y Fw(j)785 3902 y Fz(;)14 b(j)49 b Fy(2)c Fx(N)t Fy(g)h FA(a)40 b(Hilb)r(ert)h(basis)e(of)i(the)f(space) g Fz(H)2318 3872 y Fv(1)p Fw(=)p Fv(2)2422 3902 y FA(\()p Fx(T)2510 3872 y Fw(n)2555 3902 y FA(\),)k(formed)c(b)n(y)g(prop)r (erly)515 4002 y(normalised)31 b(and)i(en)n(umerated)f(non-zero)f (functions)i(sin)14 b Fz(s)k Fy(\001)h Fz(x)33 b FA(and)g(cos)13 b Fz(s)18 b Fy(\001)h Fz(x)33 b FA(\()p Fz(s)f Fy(2)g Fx(Z)3284 3972 y Fw(n)3324 4002 y FA(\),)515 4102 y(w)n(e)27 b(set)1227 4201 y Fq(e)-57 b Fz(')1270 4213 y Fw(j)1328 4201 y FA(=)23 b(\()p Fz( )1502 4213 y Fw(j)1537 4201 y Fz(;)14 b FA(0\))p Fz(;)108 b Fq(e)-57 b Fz(')1822 4213 y Fu(\000)p Fw(j)1932 4201 y FA(=)23 b(\(0)p Fz(;)14 b( )2185 4213 y Fw(j)2219 4201 y FA(\))p Fz(;)181 b(j)28 b Fy(2)23 b Fx(N)t Fz(:)515 4349 y FA(The)31 b(obtained)g(symplectic)h (scale)e(\()p Fy(f)p Fz(Z)1780 4361 y Fw(s)1815 4349 y Fy(g)p Fz(;)14 b(\013)1947 4361 y Fv(2)1984 4349 y FA(\))32 b(is)f(a)g(Darb)r(oux)g(scale.)48 b(It)31 b(is)h(easy)e(to)h (see)515 4449 y(that)d(\(8.11\))e(is)i(a)f(Hamiltonian)h(equation)f (with)h(the)g(hamiltonian)1105 4672 y Fz(H)7 b FA(\()p Fz(u;)14 b(v)s FA(\))23 b(=)1493 4616 y(1)p 1493 4653 42 4 v 1493 4729 a(2)1545 4672 y Fy(h)p Fz(B)t FA(\()p Fz(u;)14 b(v)s FA(\))p Fz(;)g FA(\()p Fz(u;)g(v)s FA(\))p Fy(i)2097 4684 y Fv(0)2153 4672 y FA(+)2236 4559 y Fq(Z)2333 4672 y Fz(F)e FA(\()p Fz(u)p FA(;)i Fz(t;)g(x)p FA(\))g Fz(dx;)515 4907 y FA(where)38 b Fz(F)831 4876 y Fu(0)819 4927 y Fw(u)905 4907 y FA(=)k Fz(f)9 b FA(.)71 b(So)38 b Fz(Z)1339 4919 y Fv(0)1418 4907 y FA(=)k Fz(H)1601 4876 y Fv(1)p Fw(=)p Fv(2)1705 4907 y FA(\()p Fx(T)1793 4876 y Fw(n)1838 4907 y Fz(;)14 b Fx(R)1929 4876 y Fv(2)1972 4907 y FA(\))40 b(is)f(the)g(Darb)r(oux)f(phase)h(space)f(for)h(the)515 5006 y(nonlinear)26 b(w)n(a)n(v)n(e)g(equation,)h(written)h(in)g(the)g (form)f(\(8.11\))o(.)1905 5255 y(31)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 32 31 bop 639 523 a FA(T)-7 b(o)27 b(apply)g(Theorem)g(8.1)f(w)n(e)h (ha)n(v)n(e)g(to)g(c)n(hec)n(k)f(the)i(conditions)f(\(H1\))h(and)f (\(H2\).)37 b(The)515 623 y(\014rst)24 b(one)g(\(with)h Fz(A)e FA(=)g Fz(B)t FA(\))i(holds)f(trivially)g(since)35 b Fq(e)-57 b Fz(')2126 635 y Fw(j)2161 623 y FA('s)24 b(are)g(eigenfunctions)g(of)g(the)h(Lapla-)515 722 y(cian.)36 b(The)28 b(condition)f(\(H2\))h(holds)g(in)f(the)h(follo)n(wing)f (three)g(cases:)639 822 y(a\))h Fz(n)23 b FA(=)f(1,)639 922 y(b\))28 b Fz(n)23 b FA(=)g(2,)k Fz(D)e Fy(\024)e FA(4,)639 1021 y(c\))28 b Fz(n)23 b FA(=)g(3,)k Fz(D)e Fy(\024)e FA(2.)639 1121 y(The)33 b(case)f(a\))g(and)h(the)g(case)e (b\))i(with)h Fz(D)f Fy(\024)e FA(2)h(can)g(b)r(e)h(c)n(hec)n(k)n(ed)f (using)g(elemen)n(tary)515 1220 y(to)r(ols,)26 b(see)f([Kuk95)o(].)36 b(Argumen)n(ts)25 b(in)h(the)h(case)e(b\))h(with)g(3)d Fy(\024)g Fz(D)i Fy(\024)d FA(4)k(and)f(in)i(the)f(case)f(c\))515 1320 y(are)h(based)h(on)h(a)f(Stric)n(hartz-t)n(yp)r(e)f(inequalit)n(y) -7 b(,)28 b(see)f([Bou95a)n(].)639 1420 y(In)e(the)g(cases)e(a\){c\),)i (Theorem)e(8.1)h(applies)g(to)g(equation)g(\(8.10\))f(in)i(the)g(form)f (\(8.11\))515 1519 y(and)32 b(sho)n(ws)f(that)i(the)g(\015o)n(w)f(maps) g(cannot)g(squeeze)g Fz(H)2313 1489 y Fv(1)p Fw(=)p Fv(2)2417 1519 y FA(-balls)g(to)g(narro)n(w)f(cylinders.)515 1619 y(This)c(result)h(can)f(b)r(e)h(in)n(terpreted)f(as)g(imp)r(ossibilit)n (y)h(of)g(`lo)r(cally)f(uniform')g(energy)g(tran-)515 1719 y(sition)g(to)h(high)f(mo)r(des,)h(see)f(in)h([Kuk95)n(].)515 1851 y Fp(Example)37 b FA(8.5)p Fp(.)k FA(F)-7 b(or)27 b(a)g(nonlinear)f(Sc)n(hr\177)-42 b(odinger)26 b(equation)1355 2033 y(_)-38 b Fz(u)23 b FA(=)g Fz(i)p FA(\001)p Fz(u)17 b FA(+)h Fz(if)1824 1999 y Fu(0)1815 2053 y Fw(u)1858 2033 y FA(\()p Fy(j)p Fz(u)p Fy(j)1984 1999 y Fv(2)2021 2033 y FA(\))p Fz(u;)180 b(x)23 b Fy(2)h Fx(T)2509 1999 y Fw(n)3167 2033 y FA(\(8.12\))515 2215 y(\(cf.)37 b(Example)24 b(2.7\),)i(the)g(Darb)r(oux)f(phase)g(space)g(is)g(the)h Fz(L)2411 2227 y Fv(2)2448 2215 y FA(-space)e Fz(L)2752 2227 y Fv(2)2789 2215 y FA(\()p Fx(T)2877 2185 y Fw(n)2922 2215 y FA(;)14 b Fx(C)h FA(\))32 b(with)26 b(the)515 2314 y(basis,)d(formed)f(b)n(y)h(normalised)e(exp)r(onen)n(ts)i Fy(f)p Fz(e)2005 2284 y Fw(is)p Fu(\001)p Fw(x)2120 2314 y Fz(;)14 b(ie)2225 2284 y Fw(is)p Fu(\001)p Fw(x)2341 2314 y Fy(g)p FA(.)35 b(It)23 b(is)f(v)n(ery)g(unlik)n(ely)h(that)g (the)515 2414 y(\015o)n(w-maps)31 b(of)40 b(\(8.12\))32 b(satisfy)g(in)i(this)f(space)f(assumption)h(\(H2\).)53 b(So)33 b(w)n(e)f(smo)r(oth)h(out)515 2514 y(the)28 b(hamiltonian)f(of) 34 b(\(8.12\))27 b(and)g(replace)g(it)h(b)n(y)1108 2738 y Fz(H)1177 2750 y Fw(\030)1237 2738 y FA(=)1335 2682 y(1)p 1335 2719 42 4 v 1335 2795 a(2)1400 2625 y Fq(Z)1483 2738 y FA(\()p Fy(jr)p Fz(u)p Fy(j)1678 2703 y Fv(2)1734 2738 y FA(+)18 b Fz(f)9 b FA(\()p Fy(j)p Fz(U)g Fy(j)2011 2703 y Fv(2)2048 2738 y FA(\)\))14 b Fz(dx;)181 b(U)31 b FA(=)23 b Fz(u)18 b Fy(\003)g Fz(\030)t(;)515 2962 y FA(where)28 b Fz(u)19 b Fy(\003)g Fz(\030)34 b FA(is)29 b(the)g(con)n(v)n(olution)f(of)h Fz(u)f FA(with)i(a)f(function)g Fz(\030)h Fy(2)c Fz(C)2602 2932 y Fu(1)2672 2962 y FA(\()p Fx(T)2760 2932 y Fw(n)2805 2962 y Fz(;)14 b Fx(R)p FA(\).)48 b(The)29 b(corre-)515 3062 y(sp)r(onding)e(Hamiltonian)g(equation)g(is) 1465 3243 y(_)-37 b Fz(u)22 b FA(=)h Fz(i)p FA(\001)p Fz(u)18 b FA(+)g Fz(i)p FA(\()p Fz(f)1967 3209 y Fu(0)1990 3243 y FA(\()p Fy(j)p Fz(U)9 b Fy(j)2134 3209 y Fv(2)2171 3243 y FA(\))p Fz(U)g FA(\))19 b Fy(\003)f Fz(\030)t(:)724 b FA(\(8.13\))515 3425 y(This)38 b(smo)r(othed)h(equation)f (satis\014es)g(\(H1\),)k(\(H2\))d(and)g(Theorem)f(8.1)g(applies)g(to)h (its)515 3525 y(\015o)n(w-maps.)515 3757 y Fs(8.4)112 b(Symplectic)35 b(capacit)m(y)515 3910 y FA(Another)c(w)n(a)n(y)f(to)h (pro)n(v)n(e)e(Theorem)h(8.1)h(uses)g(a)f(new)i(ob)5 b(ject)30 b({)h(symplectic)g(capacit)n(y)f({)515 4010 y(whic)n(h)d(is)h(in)n(teresting)f(on)g(its)h(o)n(wn.)639 4110 y(Symplectic)35 b(capacit)n(y)e(in)i(a)f(Hilb)r(ert)h(Darb)r(oux)e (space)h(\()p Fz(Z)2544 4122 y Fw(d)2583 4110 y Fz(;)14 b(\013)2673 4122 y Fv(2)2710 4110 y FA(\))35 b(as)e(in)i(section)f(8.2) 515 4209 y(\(b)r(elo)n(w)22 b(w)n(e)f(abbreviate)g Fz(Z)1353 4221 y Fw(d)1413 4209 y FA(to)h Fz(Z)6 b FA(\),)23 b(is)f(a)g(map)f Fz(c)h FA(whic)n(h)g(corresp)r(onds)e(to)i(an)n(y)f(op)r(en)h(subset) 515 4309 y Fz(O)j Fy(\032)e Fz(Z)33 b FA(a)28 b(n)n(um)n(b)r(er)f Fz(c)p FA(\()p Fz(O)r FA(\))d Fy(2)f FA([0)p Fz(;)14 b Fy(1)p FA(])28 b(and)f(satis\014es)g(the)h(follo)n(wing)e(prop)r (erties:)639 4408 y(1\))i Fp(tr)l(anslational)i(invarianc)l(e)6 b FA(:)39 b Fz(c)p FA(\()p Fz(O)r FA(\))24 b(=)f Fz(c)p FA(\()p Fz(O)e FA(+)d Fz(\030)t FA(\))28 b(for)f(an)n(y)g Fz(\030)g Fy(2)d Fz(Z)6 b FA(;)639 4508 y(2\))28 b Fp(monotonicity)7 b FA(:)38 b(if)28 b Fz(O)1420 4520 y Fv(1)1481 4508 y Fy(\033)23 b Fz(O)1632 4520 y Fv(2)1669 4508 y FA(,)28 b(then)g Fz(c)p FA(\()p Fz(O)2040 4520 y Fv(1)2078 4508 y FA(\))23 b Fy(\025)g Fz(c)p FA(\()p Fz(O)2352 4520 y Fv(2)2390 4508 y FA(\);)639 4608 y(3\))28 b(2)p Fp(-homo)l(geneity)7 b FA(:)38 b Fz(c)p FA(\()p Fz(\034)9 b(O)r FA(\))25 b(=)d Fz(\034)1692 4578 y Fv(2)1730 4608 y Fz(c)p FA(\()p Fz(O)r FA(\);)639 4707 y(4\))28 b Fp(normalisation)6 b FA(:)40 b(for)27 b(an)n(y)g(ball)h Fz(B)1817 4719 y Fw(r)1877 4707 y FA(=)23 b Fz(B)2028 4719 y Fw(r)2065 4707 y FA(\()p Fz(x)p FA(;)14 b Fz(Z)6 b FA(\))28 b(and)g(an)n(y)f(cylinder)h Fz(C)3005 4677 y Fw(j)2999 4728 y(r)3063 4707 y FA(=)c Fz(C)3217 4677 y Fw(j)3211 4728 y(r)3252 4707 y FA(\()p Fz(Z)6 b FA(\))515 4807 y(w)n(e)27 b(ha)n(v)n(e)f Fz(c)p FA(\()p Fz(B)959 4819 y Fw(r)996 4807 y FA(\))e(=)e Fz(c)p FA(\()p Fz(C)1272 4777 y Fw(j)1266 4827 y(r)1308 4807 y FA(\))h(=)g Fz(\031)s(r)1540 4777 y Fv(2)1578 4807 y Fz(:)639 4907 y FA(\(W)-7 b(e)28 b(note)g(that)g(for)f Fz(x)c FA(=)g(0)k(the)h(cylinder)f(con)n(tains)g(the)h(ball)f(and)g(is) h(`m)n(uc)n(h)f(bigger',)515 5006 y(but)h(b)r(oth)g(sets)f(ha)n(v)n(e)g (the)h(same)f(capacit)n(y)-7 b(.\))1905 5255 y(32)p eop PStoPSsaved restore %%Page: (32,33) 17 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 33 32 bop 639 523 a FA(5\))32 b Fp(Symple)l(ctic)j(invarianc)l(e)6 b FA(:)48 b(for)32 b(an)n(y)g(symplectomorphism)f(\010)9 b(:)29 b Fz(Z)37 b Fy(!)31 b Fz(Z)38 b FA(and)32 b(an)n(y)515 623 y(domain)27 b Fz(O)r FA(,)h Fz(c)p FA(\(\010\()p Fz(O)r FA(\)\))d(=)e Fz(c)p FA(\()p Fz(O)r FA(\).)639 722 y(If)i(\()p Fz(Z)q(;)14 b(\013)899 734 y Fv(2)936 722 y FA(\))25 b(is)f(a)f(\014nite-dimensional)h(Darb)r(oux)f(space,)h (then)h(existence)f(of)g(a)f(capacit)n(y)515 822 y(with)f(prop)r (erties)e(1\){5\))h(is)h(equiv)-5 b(alen)n(t)21 b(to)g(the)h(Gromo)n(v) e(theorem.)35 b(Indeed,)23 b(if)f(a)f(capacit)n(y)515 922 y(exists,)32 b(then)g(the)f(squeezing)h(\(8.4\))f(with)h Fz(\032)e(<)f(r)34 b FA(is)d(imp)r(ossible)g(due)h(to)f(2\),)i(4\))e (and)g(5\).)515 1021 y(On)c(the)h(opp)r(osite,)f(the)h(quan)n(tit)n(y) 602 1199 y(~)-43 b Fz(c)p FA(\()p Fz(O)r FA(\))24 b(=)e(sup)p Fy(f)p Fz(\031)s(r)1133 1165 y Fv(2)1194 1199 y Fy(j)h FA(there)28 b(exists)f(a)g(symplectomorphism)g(whic)n(h)g(sends)h Fz(B)3025 1211 y Fw(r)3089 1199 y FA(in)g Fz(O)s Fy(g)515 1377 y FA(ob)n(viously)39 b(satis\014es)i(1\){3\))f(and)h(5\).)77 b(Using)41 b(the)g(Gromo)n(v)f(theorem)g(w)n(e)h(see)g(that)i(~)-44 b Fz(c)515 1477 y FA(satis\014es)27 b(4\))g(as)g(w)n(ell.)639 1577 y(If)e(\()p Fz(Z)q(;)14 b(\013)899 1589 y Fv(2)937 1577 y FA(\))24 b(is)h(a)e(Hilb)r(ert)i(Darb)r(oux)f(space,)g(then)h (the)g(\014nite-dimensional)f(symplectic)515 1676 y(capacit)n(y)-7 b(,)24 b(obtained)g(in)h([HZ94)o(],)h(can)e(b)r(e)h(used)g(to)f (construct)g(a)g(capacit)n(y)g Fz(c)g FA(whic)n(h)h(meets)515 1776 y(1\){4\))o(.)34 b(This)18 b(capacit)n(y)g(turns)h(out)f(to)h(b)r (e)g(in)n(v)-5 b(arian)n(t)17 b(under)i(symplectomorphisms,)g(whic)n(h) 515 1875 y(are)39 b(\015o)n(w-maps)f Fz(S)1124 1845 y Fw(T)1119 1896 y(t)1216 1875 y FA(as)h(in)h(Theorem)f(8.1,)j(see)e ([Kuk95)n(].)74 b(This)40 b(result)f(also)g(implies)515 1975 y(Theorem)27 b(8.1.)515 2249 y FC(9)134 b(The)45 b(squeezing)g(phenomenon)g(and)716 2398 y(the)h(essen)l(tial)h(part)e (of)g(the)g(phase-space)515 2580 y FA(Example)23 b(8.4)g(sho)n(ws)g (that)i(\015o)n(w-maps)d(of)i(the)h(nonlinear)e(w)n(a)n(v)n(e)g (equation)g(\(8.11\))g(satisfy)515 2680 y(the)29 b(Gromo)n(v)f(prop)r (ert)n(y)-7 b(.)40 b(This)29 b(means)f(\(more)h(or)f(less\))h(that)g Fp(\015ow)j(of)g(gener)l(alise)l(d)g(solu-)515 2779 y(tions)d(for)i(a)f (nonline)l(ar)g(wave)g(e)l(quation)g(c)l(annot)f(sque)l(eze)h(a)f(b)l (al)t(l)i(in)f(a)f(narr)l(ow)h(cylinder)p FA(.)515 2879 y(On)37 b(the)i(con)n(trary)-7 b(,)38 b(b)r(eha)n(viour)f(of)g(the)i (\015o)n(w)e(formed)g(b)n(y)h Fp(classic)l(al)48 b FA(solutions)37 b(for)h(the)515 2979 y(nonlinear)26 b(w)n(a)n(v)n(e)f(equation)h(in)i (su\016cien)n(tly)e(smo)r(oth)h(Sob)r(olev)g(spaces)e(exhibits)j(`a)e (lot)h(of)515 3078 y(squeezing',)c(at)g(least)g(if)h(w)n(e)f(put)g(a)g (small)g(parameter)f Fz(\016)k FA(in)e(fron)n(t)e(of)i(the)f (Laplacian.)34 b(Cor-)515 3178 y(resp)r(onding)27 b(results)g(apply)h (to)g(a)f(bigger)g(class)g(of)h(equations.)37 b(Belo)n(w)27 b(w)n(e)h(discuss)g(them)515 3277 y(for)37 b(nonlinear)g(Sc)n(hr\177) -42 b(odinger)35 b(equations;)42 b(concerning)37 b(the)h(nonlinear)f(w) n(a)n(v)n(e)f(equation)515 3377 y(\(8.10\))27 b(see)g(the)h(author's)e (pap)r(er)h(in)h(GAF)-9 b(A)29 b(5:4.)639 3477 y(Let)f(us)g(consider)e (the)i(nonlinear)f(Sc)n(hr\177)-42 b(odinger)25 b(equation:)1574 3655 y(_)-37 b Fz(u)22 b FA(=)h Fy(\000)p Fz(i\016)s FA(\001)p Fz(u)17 b FA(+)i Fz(i)p Fy(j)p Fz(u)p Fy(j)2193 3620 y Fv(2)p Fw(p)2263 3655 y Fz(u;)874 b FA(\(9.1\))515 3833 y(where)23 b Fz(\016)j(>)c FA(0)h(and)g Fz(p)g Fy(2)h Fx(N)t FA(,)30 b(supplemen)n(ted)24 b(b)n(y)f(the)g(o)r(dd)h(p)r(erio)r (dic)f(b)r(oundary)f(conditions:)984 4005 y Fz(u)p FA(\()p Fz(t;)14 b(x)p FA(\))24 b(=)e Fz(u)p FA(\()p Fz(t;)14 b(x)1515 4017 y Fv(1)1553 4005 y Fz(;)g(:)g(:)g(:)f(;)h(x)1784 4017 y Fw(j)1838 4005 y FA(+)k(2)p Fz(\031)s(;)c(:)g(:)g(:)f(;)h(x)2244 4017 y Fw(n)2290 4005 y FA(\))1234 4130 y(=)22 b Fy(\000)p Fz(u)p FA(\()p Fz(t;)14 b(x)1580 4142 y Fv(1)1617 4130 y Fz(;)g(:)g(:)g(:)g(;)g Fy(\000)p Fz(x)1914 4142 y Fw(j)1949 4130 y Fz(;)g(:)g(:)g(:)f(;)h(x)2180 4142 y Fw(n)2226 4130 y FA(\))p Fz(;)180 b(j)28 b FA(=)23 b(1)p Fz(;)14 b(:)g(:)g(:)f(;)h(n;)3208 4069 y FA(\(9.2\))515 4309 y(where)19 b Fz(n)k Fy(\024)f FA(3.)34 b(Clearly)-7 b(,)20 b(an)n(y)f(function)h(whic)n(h)f(satis\014es)g(\(9.2\))g(v)-5 b(anishes)19 b(at)g(the)h(b)r(oundary)515 4408 y(of)39 b(the)h(cub)r(e)g Fz(K)1061 4378 y Fw(n)1145 4408 y FA(of)f(half-p)r (erio)r(ds,)j Fz(K)1821 4378 y Fw(n)1908 4408 y FA(=)g Fy(f)p FA(0)g Fy(\024)g Fz(x)2295 4420 y Fw(j)2374 4408 y Fy(\024)g Fz(\031)s Fy(g)p FA(.)72 b(The)39 b(problem)g(\(9.1\),)515 4508 y(\(9.2\))26 b(can)g(b)r(e)h(written)f(in)h(the)g(Hamiltonian)f (form)g(\(2.2\))g(if)h(for)f(the)h(symplectic)f(Hilb)r(ert)515 4608 y(scale)33 b(\()p Fy(f)p Fz(X)863 4620 y Fw(s)898 4608 y Fy(g)p Fz(;)14 b(\013)1030 4620 y Fv(2)1067 4608 y FA(\))34 b(one)f(tak)n(es)g(the)h(scale)f(formed)h(b)n(y)g(o)r(dd)f (p)r(erio)r(dic)h(complex)g(Sob)r(olev)515 4707 y(functions,)28 b Fz(X)965 4719 y Fw(s)1023 4707 y FA(=)23 b Fz(H)1187 4677 y Fw(s)1180 4731 y Fv(o)r(dd)1292 4707 y FA(\()p Fx(R)1378 4677 y Fw(n)1416 4707 y Fz(=)p FA(2)p Fz(\031)s Fx(Z)1610 4677 y Fw(n)1650 4707 y FA(;)14 b Fx(C)g FA(\))q(,)33 b(and)28 b Fz(\013)2044 4719 y Fv(2)2104 4707 y FA(=)23 b Fz(i)14 b(du)k Fy(^)g Fz(du)28 b FA(\(cf.)g(Example)f(2.8\).)639 4807 y(Due)20 b(to)g(a)f(non)n(trivial)f(result)h(of)h(Bourgain)d (\(whic)n(h)j(can)f(b)r(e)h(extracted)f(from)g([Bou93)o(]\),)515 4907 y(\015o)n(w-maps)24 b Fz(S)959 4876 y Fw(t)1014 4907 y FA(for)h(\(9.1\),)h(\(9.2\))g(are)f(w)n(ell)g(de\014ned)i(in)f (the)g(spaces)f Fz(X)2702 4919 y Fw(s)2737 4907 y FA(,)h Fz(s)d Fy(\025)g FA(1.)36 b(In)26 b(partic-)515 5006 y(ular,)31 b(they)g(are)e(w)n(ell)i(de\014ned)g(in)g(the)g(space)f Fz(C)2041 4976 y Fu(1)2143 5006 y FA(of)h(smo)r(oth)f(o)r(dd)h(p)r (erio)r(dic)g(functions.)1905 5255 y(33)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 34 33 bop 515 523 a FA(Denoting)20 b(b)n(y)g Fy(j)f Fr(\001)e Fy(j)1083 535 y Fw(m)1167 523 y FA(the)k Fz(C)1368 493 y Fw(m)1431 523 y FA(-norm,)g Fy(j)p Fz(u)p Fy(j)1787 535 y Fw(m)1873 523 y FA(=)i(sup)2086 543 y Fu(j)p Fw(\013)p Fu(j)p Fv(=)p Fw(m)2296 523 y FA(sup)2421 543 y Fw(x)2477 523 y Fy(j)p Fz(@)2549 493 y Fw(\013)2544 544 y(x)2596 523 y Fz(u)p FA(\()p Fz(x)p FA(\))p Fy(j)p FA(,)g(w)n(e)d(de\014ne)g(b) r(elo)n(w)515 623 y(the)32 b(set)h Fc(A)857 635 y Fw(m)951 623 y Fy(\032)d Fz(C)1111 593 y Fu(1)1214 623 y FA(whic)n(h)i(w)n(e)g (call)g(the)h(essen)n(tial)f(part)g(of)g(the)h(smo)r(oth)f(phase-space) 515 722 y(for)c(the)i(problem)f(\(9.1\))o(,)h(\(9.2\))f(with)g(resp)r (ect)g(to)g(the)h Fz(C)2325 692 y Fw(m)2388 722 y FA(-norm,)f(or)f (just)i(the)f Fp(essential)515 822 y(p)l(art)h(of)g(the)g(phase-sp)l (ac)l(e)6 b FA(:)891 1005 y Fc(A)951 1017 y Fw(m)1037 1005 y FA(=)22 b Fy(f)p Fz(u)g Fy(2)i Fz(C)1380 970 y Fu(1)1473 1005 y Fy(j)g Fz(u)j FA(satis\014es)g(\(9.2\))g(and)g(the)h (condition)g(\(9.3\))o Fy(g)p Fz(;)515 1187 y FA(where)1468 1287 y Fy(j)p Fz(u)p Fy(j)1562 1299 y Fv(0)1622 1287 y Fy(\024)23 b Fz(K)1781 1299 y Fw(m)1843 1287 y Fz(\016)1883 1253 y Fw(\026)1928 1287 y Fy(j)p Fz(u)p Fy(j)2022 1253 y Fv(1)p Fw(=)p Fv(\(2)p Fw(pm)p Fo({)s Fv(+1\))2022 1307 y Fw(m)2402 1287 y Fz(;)783 b FA(\(9.3\))515 1436 y(with)33 b(a)e(suitable)i Fz(K)1168 1448 y Fw(m)1261 1436 y FA(=)e Fz(K)1428 1448 y Fw(m)1490 1436 y FA(\()p Fx({)s FA(\))j(and)e Fz(\026)f FA(=)f Fz(m)p Fx({)s Fz(=)p FA(\(2)p Fz(pm)p Fx({)25 b FA(+)c(1\).)51 b(Here)32 b Fx({)k FA(is)d(an)n(y)e(\014xed)515 1536 y(constan)n(t)c Fx({)f Fy(2)e FA(\(0)p Fz(;)14 b FA(1)p Fz(=)p FA(3\).)639 1636 y(In)n(tersection)25 b(of)i(the)f(set)g Fc(A)1513 1648 y Fw(m)1602 1636 y FA(with)h(the)f Fz(R)q FA(-sphere)f(in)h(the)h Fz(C)2582 1605 y Fw(m)2645 1636 y FA(-norm)e(\(i.e.,)i(with)g(the)515 1735 y(set)34 b Fy(fj)p Fz(u)p Fy(j)787 1747 y Fw(m)884 1735 y FA(=)g Fz(R)q Fy(g)p FA(\))g(has)g(the)h Fz(C)1525 1705 y Fv(0)1563 1735 y FA(-diameter)e Fy(\024)h FA(2)p Fz(K)2155 1747 y Fw(m)2218 1735 y Fz(\016)2258 1705 y Fw(\026)2302 1735 y Fz(R)2366 1705 y Fv(1)p Fw(=)p Fv(\(2)p Fw(pm)p Fo({)s Fv(+1\))2747 1735 y FA(.)57 b(Asymptotically)515 1835 y(\(as)27 b Fz(\016)f Fy(!)d FA(0)k(or)f Fz(R)e Fy(!)f(1)p FA(\))28 b(this)f(is)g(m)n(uc)n(h)g(smaller)g(than)g(the)h Fz(C)2477 1805 y Fv(0)2514 1835 y FA(-diameter)f(of)g(the)h(sphere,)515 1934 y(whic)n(h)19 b(equals)f Fz(C)1046 1946 y Fw(m)1109 1934 y Fz(R)q FA(.)34 b(Th)n(us,)21 b Fc(A)1517 1946 y Fw(m)1598 1934 y FA(is)e(an)g(`asymptotically)f(narro)n(w')f(subset)i (of)g(the)g(smo)r(oth)515 2034 y(phase)27 b(space.)639 2134 y(The)37 b(theorem)g(b)r(elo)n(w)g(states)f(that)h(for)g(an)n(y)f Fz(m)j Fy(\025)f FA(2)f(the)g(set)g Fc(A)2779 2146 y Fw(m)2879 2134 y FA(is)g(a)g(recursion)515 2233 y(subset)27 b(for)g(the)h(dynamical)f(system,)h(and)f(giv)n(es)g(a)g(con)n(trol)f (for)h(the)h(recursion)e(time:)515 2399 y FB(Theorem)42 b(9.1.)k Fp(L)l(et)39 b Fz(u)p FA(\()p Fz(t)p FA(\))h(=)g Fz(u)p FA(\()p Fz(t;)28 b Fr(\001)13 b FA(\))40 b Fp(b)l(e)f(a)h(smo)l (oth)f(solution)h(for)49 b FA(\(9.1\))p Fp(,)42 b FA(\(9.2\))c Fp(and)515 2499 y Fy(j)p Fz(u)p FA(\()p Fz(t)648 2511 y Fv(0)685 2499 y FA(\))p Fy(j)740 2511 y Fv(0)800 2499 y FA(=)23 b Fz(U)9 b Fp(.)38 b(Then)28 b(ther)l(e)f(exists)g Fz(T)35 b Fy(\024)22 b Fz(t)1861 2511 y Fv(0)1912 2499 y FA(+)13 b Fz(\016)2030 2469 y Fu(\000)p Fv(1)p Fw(=)p Fv(3)2186 2499 y Fz(U)2252 2469 y Fu(\000)p Fv(4)p Fw(p=)p Fv(3)2470 2499 y Fp(such)27 b(that)h Fz(u)p FA(\()p Fz(T)12 b FA(\))22 b Fy(2)h Fc(A)3157 2511 y Fw(m)3247 2499 y Fp(and)525 2566 y Fv(1)p 525 2580 34 4 v 525 2627 a(2)568 2599 y Fz(U)32 b Fy(\024)22 b(j)p Fz(u)p FA(\()p Fz(T)12 b FA(\))p Fy(j)963 2611 y Fv(0)1023 2599 y Fy(\024)1120 2566 y Fv(3)p 1120 2580 V 1120 2627 a(2)1163 2599 y Fz(U)d Fp(.)639 2765 y FA(Since)29 b Fz(L)914 2777 y Fv(2)951 2765 y FA(-norm)f(of)h(a)g(solution)f(is)h(an)g(in)n(tegral)f(of)h (motion)g(\(see)g(Example)f(3.5\))g(and)515 2864 y Fy(j)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(j)703 2876 y Fv(0)763 2864 y Fy(\025)23 b(j)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(j)1039 2879 y Fw(L)1085 2887 y Ft(2)1117 2879 y Fv(\()p Fw(K)1203 2863 y Fn(n)1244 2879 y Fv(\))1274 2864 y FA(,)k(then)h(w)n (e)g(obtain)f(the)h(follo)n(wing)515 3030 y FB(Corollary)d(9.2.)34 b Fp(L)l(et)23 b Fz(u)p FA(\()p Fz(t)p FA(\))g Fp(b)l(e)h(a)f(smo)l (oth)h(solution)g(for)33 b FA(\(9.1\))p Fp(,)25 b FA(\(9.2\))e Fp(and)h Fy(j)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(j)3057 3045 y Fw(L)3103 3053 y Ft(2)3135 3045 y Fv(\()p Fw(K)3221 3029 y Fn(n)3261 3045 y Fv(\))3314 3030 y Fy(\021)515 3130 y Fz(W)12 b Fp(.)46 b(Then)33 b(for)g(any)f Fz(m)c Fy(\025)f FA(2)32 b Fp(this)g(solution)h(c)l(annot)f(stay)g(outside)h Fc(A)2731 3142 y Fw(m)2826 3130 y Fp(longer)f(than)h(the)515 3230 y(time)d Fz(\016)744 3199 y Fu(\000)p Fv(1)p Fw(=)p Fv(3)900 3230 y Fz(W)990 3199 y Fu(\000)p Fv(4)p Fw(p=)p Fv(3)1180 3230 y Fp(.)639 3396 y FA(F)-7 b(or)26 b(the)g(theorem's)f (pro)r(of)h(w)n(e)f(refer)h(the)g(reader)f(to)h(App)r(endix)g(3)g(in)g ([Kuk99)o(].)36 b(Here)515 3495 y(w)n(e)41 b(explain)g(wh)n(y)g (`something)g(lik)n(e)g(this)h(result')f(should)h(b)r(e)f(true.)79 b(Presen)n(ting)40 b(the)515 3595 y(argumen)n(ts)c(it)i(is)g(more)f (con)n(v)n(enien)n(t)f(to)i(op)r(erate)f(with)h(the)g(Sob)r(olev)f (norms)g Fy(k)i Fr(\001)f Fy(k)3293 3607 y Fw(m)3356 3595 y FA(.)515 3694 y(Let)30 b(us)g(denote)g Fy(k)p Fz(u)p FA(\()p Fz(t)1197 3706 y Fv(0)1233 3694 y FA(\))p Fy(k)1307 3706 y Fv(0)1372 3694 y FA(=)c Fz(A)p FA(.)45 b(Arguing)29 b(b)n(y)h(con)n(tradiction,)f(w)n(e)h(assume)f(that)i(for) e(all)515 3794 y Fz(t)23 b Fy(2)g FA([)p Fz(t)699 3806 y Fv(0)736 3794 y Fz(;)14 b(t)803 3806 y Fv(1)841 3794 y FA(])23 b(=)f Fz(L)p FA(,)28 b(where)f Fz(t)1352 3806 y Fv(1)1412 3794 y FA(=)c Fz(t)1530 3806 y Fv(0)1585 3794 y FA(+)18 b Fz(\016)1708 3764 y Fu(\000)p Fv(1)p Fw(=)p Fv(3)1865 3794 y Fz(U)1931 3764 y Fu(\000)p Fv(4)p Fw(p=)p Fv(3)2121 3794 y FA(,)28 b(w)n(e)f(ha)n(v)n(e)1627 3977 y Fz(C)6 b(\016)1732 3942 y Fw(a)1772 3977 y Fy(k)p Fz(u)p Fy(k)1904 3942 y Fw(b)1904 3997 y(m)1989 3977 y Fz(<)22 b Fy(k)p Fz(u)p Fy(k)2208 3989 y Fv(0)2244 3977 y Fz(;)941 b FA(\(9.4\))515 4159 y(where)27 b Fz(m)c Fy(\025)g FA(3)k(is)h(a)f(\014xed)h(n)n(um)n(b)r(er.)37 b(Since)28 b Fy(k)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(k)2140 4171 y Fv(0)2199 4159 y Fy(\021)23 b Fz(A)p FA(,)28 b(then)h(\(9.4\))e (and)h(the)g(in)n(terp)r(o-)515 4259 y(lation)f(inequalit)n(y)g(imply)h (the)g(upp)r(er)g(b)r(ounds)1049 4452 y Fy(k)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(k)1275 4464 y Fw(l)1323 4452 y Fy(\024)23 b Fz(C)1470 4464 y Fw(l)1495 4452 y Fz(A)1557 4418 y Fv(1)p Fu(\000)1668 4395 y Fn(l)p 1653 4404 51 3 v 1653 4438 a(m)1714 4418 y Fv(+)1803 4395 y Fn(l)p 1774 4404 78 3 v 1774 4438 a(mb)1880 4452 y Fz(\016)1920 4418 y Fu(\000)1994 4395 y Fn(la)p 1982 4404 V 1982 4438 a(mb)2073 4452 y Fz(;)97 b FA(0)22 b Fy(\024)h Fz(l)i Fy(\024)d Fz(m;)42 b(t)23 b Fy(2)g Fz(L)14 b FA(;)363 b(\(9.5\))515 4643 y(In)33 b(particular,)67 b Fz(\016)s Fy(k)p FA(\001)p Fz(u)p Fy(k)1312 4655 y Fv(1)1380 4643 y Fy(\024)32 b Fz(C)1536 4655 y Fv(3)1573 4643 y Fz(A)1635 4613 y Fv(1)p Fu(\000)1742 4591 y Ft(3)p 1731 4600 51 3 v 1731 4633 a Fn(m)1792 4613 y Fv(+)1877 4591 y Ft(3)p 1853 4600 78 3 v 1853 4633 a Fn(mb)1958 4643 y Fz(\016)1998 4613 y Fv(1)p Fu(\000)2101 4591 y Ft(3)p Fn(a)p 2093 4600 V 2093 4633 a(mb)2184 4643 y Fz(:)h FA(Therefore)f(if)i Fz(mb)e(>)g FA(3)p Fz(a)p FA(,)i(then)g(for)515 4743 y Fz(t)24 b Fy(2)h Fz(L)k FA(equation)f(\(9.1\))o(,)h(treated)f(as)g(a) g(dynamical)g(system)h(in)g Fz(H)2608 4713 y Fv(1)2601 4766 y(o)r(dd)2713 4743 y FA(,)g(is)f(a)h(p)r(erturbation)515 4842 y(of)e(the)h(trivial)f(equation)1750 4942 y(_)-37 b Fz(u)22 b FA(=)h Fz(i)p Fy(j)p Fz(u)p Fy(j)2017 4908 y Fv(2)p Fw(p)2088 4942 y Fz(u:)1049 b FA(\(9.6\))1905 5255 y(34)p eop PStoPSsaved restore %%Page: (34,35) 18 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 35 34 bop 515 523 a FA(Elemen)n(tary)36 b(argumen)n(ts)g(sho)n(w)g (that)i(the)g Fz(H)2017 493 y Fv(1)2054 523 y FA(-norm)e(of)h(eac)n(h)g (non-zero)f(solution)h(for)515 623 y(\(9.6\))30 b(gro)n(ws)f(linearly)h (with)h(time.)46 b(This)31 b(implies)g(a)f(lo)n(w)n(er)f(b)r(ound)i (for)f(sup)2960 643 y Fw(t)p Fu(2)p Fw(L)3094 623 y Fy(k)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(k)3320 635 y Fv(1)3356 623 y FA(,)515 722 y(where)22 b Fz(u)p FA(\()p Fz(t)p FA(\))i(is)f(the)g(solution)g(for)g(\(9.1\))o(,)i(\(9.2\))d(whic)n(h)i (w)n(e)e(discuss.)35 b(It)24 b(turns)f(out)g(that)g(one)515 822 y(can)30 b(c)n(ho)r(ose)f Fz(a)h FA(and)h Fz(b)f FA(in)h(suc)n(h)f(a)g(w)n(a)n(y)f(that)i Fz(mb)c(>)g FA(3)p Fz(a)j FA(and)h(the)f(lo)n(w)n(er)f(b)r(ound)i(w)n(e)f(ha)n(v)n (e)515 922 y(just)f(obtained)g(con)n(tradicts)f(\(9.5\))h(with)g Fz(l)e FA(=)e(1.)41 b(This)28 b(con)n(tradiction)g(sho)n(ws)g(that)h (\(9.4\))515 1021 y(cannot)h(b)r(e)h(true)f(for)g(all)g Fz(t)e Fy(2)g Fz(L)p FA(.)46 b(In)30 b(other)g(w)n(ords,)g Fy(k)p Fz(u)p FA(\()p Fz(\034)9 b FA(\))p Fy(k)2431 1033 y Fv(0)2496 1021 y Fy(\024)28 b Fz(C)6 b(\016)2694 991 y Fw(a)2734 1021 y Fy(k)p Fz(u)p FA(\()p Fz(\034)j FA(\))p Fy(k)2975 991 y Fw(b)2975 1042 y(m)3069 1021 y FA(for)30 b(some)515 1121 y Fz(\034)k Fy(2)25 b Fz(L)p FA(.)39 b(A)n(t)29 b(this)g(momen)n(t)f Fz(\034)39 b FA(the)29 b(solution)f(en)n(ters)f(a)h(domain,)h(similar)f(to)g(the)h(essen)n (tial)515 1220 y(part)e Fc(A)755 1232 y Fw(m)817 1220 y FA(.)p 3318 1220 4 57 v 3322 1168 50 4 v 3322 1220 V 3372 1220 4 57 v 639 1370 a(Let)j(us)f(consider)f(an)n(y)h(tra)5 b(jectory)27 b Fz(u)p FA(\()p Fz(t)p FA(\))j(for)f(\(9.1\))o(,)h (\(9.2\))f(suc)n(h)g(that)h Fy(j)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(j)3054 1385 y Fw(L)3100 1393 y Ft(2)3132 1385 y Fv(\()p Fw(K)3218 1368 y Fn(n)3258 1385 y Fv(\))3314 1370 y Fy(\021)515 1481 y Fz(W)49 b Fy(\030)36 b FA(1,)i(and)d(discuss) h(the)g(time-a)n(v)n(erages)d Fy(hj)p Fz(u)p Fy(j)2107 1493 y Fw(m)2170 1481 y Fy(i)j FA(and)g Fy(hk)p Fz(u)p Fy(k)2572 1451 y Fv(2)2572 1502 y Fw(m)2634 1481 y Fy(i)2666 1451 y Fv(1)p Fw(=)p Fv(2)2807 1481 y FA(of)g(its)g Fz(C)3099 1451 y Fw(m)3162 1481 y FA(-norm)515 1581 y Fy(j)p Fz(u)p Fy(j)609 1593 y Fw(m)699 1581 y FA(and)28 b(its)f(Sob)r(olev)g(norm)g Fy(k)p Fz(u)p Fy(k)1634 1593 y Fw(m)1696 1581 y FA(,)h(where)f(w)n(e)g (set)837 1825 y Fy(hj)p Fz(u)p Fy(j)963 1837 y Fw(m)1026 1825 y Fy(i)d FA(=)1188 1769 y(1)p 1179 1806 61 4 v 1179 1882 a Fz(T)1263 1712 y Fq(Z)1346 1733 y Fw(T)1309 1901 y Fv(0)1412 1825 y Fy(j)p Fz(u)p Fy(j)1506 1837 y Fw(m)1583 1825 y Fz(dt;)180 b Fy(hk)p Fz(u)p Fy(k)2023 1791 y Fv(2)2023 1846 y Fw(m)2085 1825 y Fy(i)2117 1791 y Fv(1)p Fw(=)p Fv(2)2244 1825 y FA(=)2332 1733 y Fq(\020)2401 1769 y FA(1)p 2392 1806 V 2392 1882 a Fz(T)2476 1712 y Fq(Z)2559 1733 y Fw(T)2522 1901 y Fv(0)2625 1825 y Fy(k)p Fz(u)p Fy(k)2757 1791 y Fv(2)2806 1825 y Fz(dt)2879 1733 y Fq(\021)2929 1750 y Fv(1)p Fw(=)p Fv(2)3033 1825 y Fz(;)515 2050 y FA(and)33 b(the)g(time)g Fz(T)44 b FA(of)33 b(a)n(v)n(eraging)d(is)j (sp)r(eci\014ed)g(b)r(elo)n(w.)53 b(While)33 b(the)h(tra)5 b(jectory)31 b(sta)n(ys)h(in)515 2150 y Fc(A)575 2162 y Fw(m)637 2150 y FA(,)c(w)n(e)f(ha)n(v)n(e)1415 2249 y Fy(j)p Fz(u)p Fy(j)1509 2261 y Fw(m)1595 2249 y Fy(\025)c FA(\()p Fz(W)12 b(K)1882 2215 y Fu(\000)p Fv(1)1876 2270 y Fw(m)1971 2249 y Fz(\016)2011 2215 y Fu(\000)p Fw(\026)2107 2249 y FA(\))2139 2215 y Fv(1)p Fw(=)p Fv(\(1)p Fu(\000)p Fv(2)p Fw(p\026)p Fv(\))2455 2249 y Fz(:)515 2399 y FA(One)32 b(can)f(sho)n(w)h(that)g(this)g(inequalit)n(y)g(implies)g(that)h(eac)n (h)e(visit)h(to)g Fc(A)2817 2411 y Fw(m)2912 2399 y FA(increases)f(the) 515 2498 y(in)n(tegral)811 2431 y Fq(R)880 2498 y Fy(j)p Fz(u)p Fy(j)974 2510 y Fw(m)1051 2498 y Fz(dt)22 b FA(b)n(y)f(a)h(term) f(bigger)g(than)h Fz(\016)i FA(to)e(a)f(negativ)n(e)g(degree.)34 b(Since)22 b(these)f(visits)515 2598 y(are)31 b(su\016cien)n(tly)h (frequen)n(t)g(b)n(y)g(the)g(Corollary)-7 b(,)31 b(then)i(w)n(e)e (obtain)h(a)g(lo)n(w)n(er)f(estimate)h(for)515 2697 y(the)d(quan)n(tit) n(y)g Fy(hj)p Fz(u)p Fy(j)1119 2709 y Fw(m)1182 2697 y Fy(i)p FA(.)41 b(Details)29 b(can)g(b)r(e)h(found)f(in)g(the)h (author's)e(pap)r(er)g(in)i(CMPh)e(178,)515 2797 y(pp.)c(265{280.)33 b(Here)24 b(w)n(e)f(presen)n(t)h(a)g(b)r(etter)g(result)g(whic)n(h)g (estimates)g(the)h(time-a)n(v)n(eraged)515 2897 y(Sob)r(olev)i(norms.) 36 b(F)-7 b(or)27 b(a)g(pro)r(of)g(see)g(section)g(4.1)g(of)h([Kuk99)n (].)515 3063 y FB(Theorem)h(9.3.)40 b Fp(L)l(et)28 b Fz(u)p FA(\()p Fz(t)p FA(\))h Fp(b)l(e)g(a)h(smo)l(oth)f(solution)g (for)h(the)f(e)l(quation)36 b FA(\(9.1\))p Fp(,)29 b FA(\(9.2\))g Fp(such)515 3162 y(that)35 b Fy(j)p Fz(u)p FA(\()p Fz(t)p FA(\))p Fy(j)878 3177 y Fw(L)924 3185 y Ft(2)957 3177 y Fv(\()p Fw(K)1043 3161 y Fn(n)1083 3177 y Fv(\))1147 3162 y Fy(\025)f FA(1)p Fp(.)56 b(Then)36 b(ther)l(e)g(exists)g(a)g(se)l(quenc)l(e)f Fz(k)2502 3174 y Fw(m)2599 3162 y Fy(\045)f FA(1)p Fz(=)p FA(3)h Fp(and)h(c)l(onstants)515 3274 y Fz(C)574 3286 y Fw(m)678 3274 y Fz(>)41 b FA(0)p Fp(,)h Fz(\016)930 3286 y Fw(m)1033 3274 y Fz(>)f FA(0)e Fp(such)h(that)f Fy(hk)p Fz(u)p Fy(k)1762 3244 y Fv(2)1762 3294 y Fw(m)1824 3274 y Fy(i)1856 3244 y Fv(1)p Fw(=)p Fv(2)2002 3274 y Fy(\025)h Fz(C)2166 3286 y Fw(m)2230 3274 y Fz(\016)2270 3244 y Fu(\000)p Fv(2)p Fw(mk)2449 3252 y Fn(m)2508 3274 y Fp(,)j(pr)l(ovide)l(d)e(that) f Fz(m)h Fy(\025)f FA(4)p Fp(,)515 3374 y Fz(\016)26 b Fy(\024)c Fz(\016)702 3386 y Fw(m)795 3374 y Fp(and)30 b Fz(T)k Fy(\025)23 b Fz(\016)1167 3343 y Fu(\000)p Fv(1)p Fw(=)p Fv(3)1323 3374 y Fp(.)639 3540 y FA(The)j(results)e(stated)i(in) f(Theorems)g(9.1,)g(9.3)f(remain)h(true)g(for)g(equations)f(\(9.1\))i (with)515 3639 y(dissipation.)65 b(I.e.,)40 b(for)d(the)g(equations)g (with)g Fz(\016)k FA(replaced)36 b(b)n(y)h Fz(\016)s(\027)5 b FA(,)40 b(where)d Fz(\027)42 b FA(is)c(a)e(unit)515 3739 y(complex)26 b(n)n(um)n(b)r(er)g(suc)n(h)h(that)g(Re)14 b Fz(\027)28 b Fy(\025)23 b FA(0)j(and)g(Im)14 b Fz(\027)29 b Fy(\025)22 b FA(0.)2375 3709 y Fv(11)2481 3739 y FA(If)28 b(Im)14 b Fz(\027)28 b(>)23 b FA(0,)j(then)h(smo)r(oth)515 3838 y(solutions)i(for)g(\(9.1\),)h(\(9.2\))g(con)n(v)n(erge)e(to)h (zero)g(in)h(an)n(y)f Fz(C)2364 3808 y Fw(m)2428 3838 y FA(-norm.)43 b(Since)30 b(the)g(essen)n(tial)515 3938 y(part)j Fc(A)761 3950 y Fw(m)857 3938 y FA(clearly)g(con)n(tains)g(a)g (su\016cien)n(tly)h(small)f Fz(C)2246 3908 y Fw(m)2309 3938 y FA(-neigh)n(b)r(ourho)r(o)r(d)g(of)g(zero,)i(then)515 4038 y(ev)n(en)n(tually)i(an)n(y)h(smo)r(oth)g(solution)g(en)n(ter)g Fc(A)2000 4050 y Fw(m)2101 4038 y FA(and)h(sta)n(ys)e(there)h(forev)n (er.)68 b(Theorem)515 4137 y(9.3)29 b(states)g(that)h(the)g(solution)f (will)h(visit)g(the)g(essen)n(tial)f(part)g(m)n(uc)n(h)h(earlier,)e(b)r (efore)i(its)515 4237 y(norm)f(deca)n(ys.)43 b(Moreo)n(v)n(er,)28 b(results,)j(similar)e(to)h(Theorem)f(9.3,)h(are)f(true)h(for)g (solutions)515 4337 y(of)e(the)g(damp)r(ed-driv)n(en)g(equation)42 b(_)-38 b Fz(u)18 b FA(+)h Fz(\016)s FA(\001)p Fz(u)f Fy(\000)h Fz(i)p Fy(j)p Fz(u)p Fy(j)2195 4307 y Fv(2)2231 4337 y Fz(u)24 b FA(=)f Fz(\021)s FA(\()p Fz(t;)14 b(x)p FA(\),)29 b(where)f(the)g(force)g Fz(\021)j FA(is)515 4436 y(a)24 b(random)g(\014eld,)h(smo)r(oth)g(in)g Fz(x)p FA(,)g(and)g(stationary)e(mixing)h(in)h Fz(t)p FA(.)36 b(See)25 b(the)g(author's)f(pap)r(er)515 4536 y(in)k(GAF)-9 b(A)28 b(7,)f(783-822,)e(and)i([Kuk99)o(].)p 515 4589 1146 4 v 577 4648 a Fm(11)642 4671 y Fl(The)d(only)g(correction)g(is)f (that)i(if)e(Im)11 b Fk(\027)23 b(>)d Fl(0,)j(then)i(in)e(Theorem)g (9.3)h(one)g(should)g(tak)n(e)h Fk(T)30 b Fl(=)19 b Fk(\016)3206 4648 y Fb(\000)p Fm(1)p Fg(=)p Fm(3)3352 4671 y Fl(.)1905 5255 y FA(35)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 36 35 bop 515 523 a FC(10)135 b(App)t(endix:)66 b(F)-11 b(amilies)50 b(of)f(p)t(erio)t(dic)g(orbits)g(in)g(re-)784 672 y(v)l(ersible)d(PDEs.)60 b(By)45 b(Dario)h(Bam)l(busi)515 871 y Fs(10.1)112 b(In)m(tro)s(duction)515 1024 y FA(Some)23 b(families)g(of)f(p)r(erio)r(dic)h(solutions)g(of)g(PDEs)f(can)g(b)r(e) i(constructed)e(using)h(KAM)g(the-)515 1124 y(ory;)i(ho)n(w)n(ev)n(er)e (a)h(di\013eren)n(t)h(approac)n(h)e(leading)i(to)g(stronger)e(results)h (and)h(simpler)g(pro)r(ofs)515 1223 y(is)c(a)n(v)-5 b(ailable.)33 b(It)22 b(is)f(based)g(on)g(the)h(Ly)n(apuno)n(v{Sc)n(hmidt)e(decomp)r (osition)h(com)n(bined)g(with)515 1323 y(a)i(suitable)g(analysis)f(of)h (small)g(denominators.)34 b(The)23 b(main)g(adv)-5 b(an)n(tage)22 b(of)h(this)h(approac)n(h)515 1423 y(is)29 b(elimination)g(of)h(the)f (second)g(Melnik)n(o)n(v)f(condition)h(\(see)h(\(5.7\))o(\))g(whic)n(h) f(allo)n(ws)f(to)i(ap-)515 1522 y(ply)d(it)g(to)g(problems)f(with)i(p)r (erio)r(dic)f(b)r(oundary)f(conditions)h(and)g(to)f(equations)h(in)g (more)515 1622 y(than)k(one)g(space)f(dimension.)47 b(Most)31 b(of)g(the)g(general)f(theory)g(has)h(b)r(een)g(dev)n(elop)r(ed)g(for) 515 1722 y(equations)i(that)i(are)e(of)i(second)e(order)g(in)i(time)g (and)f(w)n(e)g(will)h(mainly)f(deal)g(with)h(this)515 1821 y(case.)k(Moreo)n(v)n(er,)26 b(w)n(e)i(will)h(concen)n(trate)e(on) h(problems)g(in)n(v)n(olving)f(small)h(denominators)515 1921 y(and)f(only)g(brie\015y)h(rep)r(ort)e(on)i(results)f(of)g(a)h (di\013eren)n(t)f(kind.)515 2151 y Fs(10.2)112 b(An)37 b(abstract)h(theorem)f(for)g(nonresonan)m(t)i(PDEs)515 2304 y FA(Let)23 b Fy(f)p Fz(X)770 2316 y Fw(s)805 2304 y Fy(g)g FA(b)r(e)h(a)f(scale)f(of)i(Hilb)r(ert)g(spaces)e(with)i (norms)f Fy(k)10 b(\001)g(k)2419 2316 y Fw(s)2477 2304 y FA(and)23 b(scalar)f(pro)r(duct)h Fy(h\001)p FA(;)14 b Fy(\001i)3320 2316 y Fw(s)3356 2304 y FA(.)515 2403 y(Let)21 b Fz(A)g FA(b)r(e)h(a)e(\(linear\))h(morphism)g(of)g(the)g (scale,)h(and)f(assume)f(that)h(there)g(exists)g(a)f(Hilb)r(ert)515 2503 y(basis)27 b Fy(f)p Fz(')815 2515 y Fw(j)849 2503 y Fy(g)891 2473 y Fu(1)891 2525 y Fw(j)s Fv(=1)1038 2503 y FA(suc)n(h)g(that)1532 2684 y Fz(A')1648 2696 y Fw(j)1707 2684 y FA(=)22 b Fz(!)1849 2650 y Fv(2)1846 2704 y Fw(j)1886 2684 y Fz(')1940 2696 y Fw(j)2003 2684 y Fz(;)97 b(!)2175 2696 y Fw(j)2233 2684 y Fz(>)22 b FA(0)515 2851 y(Let)k(us)f(\014x)h Fz(s)p FA(,)g(consider)f(a)g(neigh)n(b)r(ourho)r(o)r(d)g Fy(U)34 b FA(of)26 b(the)g(origin)f(in)h Fz(X)2643 2863 y Fw(s)2704 2851 y FA(and)f(a)h(smo)r(oth)f(map)515 2951 y Fz(g)31 b FA(:)d Fy(U)37 b(!)29 b Fz(X)907 2963 y Fw(s)942 2951 y FA(,)j(ha)n(ving)e(at)g(the)i(origin)d(a)i(zero)f(of)h(second)f (order.)45 b(W)-7 b(e)31 b(are)f(in)n(terested)h(in)515 3051 y(families)c(of)h(small)f(amplitude)h(p)r(erio)r(dic)g(solutions)e (of)i(the)g(equation)1665 3218 y(\177)-47 b Fz(x)18 b FA(+)g Fz(Ax)24 b FA(=)f Fz(g)s FA(\()p Fz(x)p FA(\))28 b Fz(:)933 b FA(\(10.1\))515 3385 y Fp(Example)32 b FA(10.1)p Fp(.)k FA(The)23 b(nonlinear)e(w)n(a)n(v)n(e)g(equation)h(with)h(p)r (erio)r(dic)g(b)r(oundary)e(conditions:)1388 3553 y Fz(w)1447 3565 y Fw(tt)1520 3553 y Fy(\000)d Fz(w)1662 3565 y Fw(xx)1761 3553 y FA(+)g Fz(V)g FA(\()p Fz(x)p FA(\))p Fz(w)27 b FA(=)c Fz(f)9 b FA(\()p Fz(x;)14 b(w)r FA(\))29 b Fz(;)661 b FA(\(10.2\))1023 3677 y Fz(w)r FA(\()p Fz(x;)14 b(t)p FA(\))25 b(=)d Fz(w)r FA(\()p Fz(x)e FA(+)e(2)p Fz(\031)s(;)c(t)p FA(\))28 b Fz(;)97 b(w)2015 3689 y Fw(x)2057 3677 y FA(\()p Fz(x;)14 b(t)p FA(\))24 b(=)f Fz(w)2406 3689 y Fw(x)2448 3677 y FA(\()p Fz(x)c FA(+)f(2)p Fz(\031)s(;)c(t)p FA(\))28 b Fz(;)296 b FA(\(10.3\))515 3845 y(where)22 b(the)i(p)r(oten)n(tial)f Fz(V)42 b FA(and)23 b(the)h(nonlinearit)n(y)e Fz(f)32 b FA(are)22 b(p)r(erio)r(dic)h(of)g(p)r(erio)r(d)g(2)p Fz(\031)j FA(in)e Fz(x)p FA(,)g(and)515 3945 y Fz(f)9 b FA(\()p Fz(x;)14 b(w)r FA(\))29 b(=)g Fz(O)r FA(\()p Fy(j)p Fz(w)r Fy(j)1101 3914 y Fv(2)1140 3945 y FA(\).)47 b(The)31 b(frequencies)f Fz(!)1896 3957 y Fw(j)1962 3945 y FA(of)g(the)i(linearised)e(system)g(are)g(the)h(square)515 4044 y(ro)r(ots)c(of)i(the)g(p)r(erio)r(dic)f(eigen)n(v)-5 b(alues)27 b(of)i(the)g(Sturm)f(Liouville)g(op)r(erator)f Fy(\000)p Fz(@)2995 4056 y Fw(xx)3093 4044 y FA(+)19 b Fz(V)g FA(\()p Fz(x)p FA(\),)515 4144 y(that)30 b(w)n(e)f(assume)g (to)h(b)r(e)g(p)r(ositiv)n(e.)44 b(In)30 b(this)g(case)f Fz(X)2193 4156 y Fw(s)2255 4144 y FA(=)d Fz(H)2422 4114 y Fw(s)2457 4144 y FA(\()p Fx(T)p FA(\).)44 b(A)30 b(smo)r(oth)g Fz(f)38 b FA(induces)515 4243 y(a)27 b(smo)r(oth)g(op)r(erator)f(from)h Fz(X)1477 4255 y Fw(s)1540 4243 y FA(to)h(itself,)g(pro)n(vided)e(that) i Fz(s)23 b(>)g FA(1)p Fz(=)p FA(2.)p 3318 4243 4 57 v 3322 4191 50 4 v 3322 4243 V 3372 4243 4 57 v 515 4370 a Fp(Example)37 b FA(10.2)p Fp(.)j FA(The)28 b(nonlinear)f(plate)g (equation)g(in)h(the)g Fz(d)g FA(dimensional)f(cub)r(e:)1310 4538 y Fz(w)1369 4550 y Fw(tt)1443 4538 y FA(+)18 b(\001\001)p Fz(w)j FA(+)d Fz(aw)26 b FA(=)c Fz(f)9 b FA(\()p Fz(w)r FA(\))29 b Fz(;)97 b(x)23 b Fy(2)h(Q)583 b FA(\(10.4\))1571 4671 y Fz(w)1632 4601 y Fq(\014)1632 4651 y(\014)1660 4705 y Fw(@)t Fu(Q)1781 4671 y FA(=)22 b(\001)p Fz(w)r Fy(j)2021 4683 y Fw(@)t Fu(Q)2143 4671 y FA(=)h(0)k Fz(;)844 b FA(\(10.5\))515 4839 y(where)27 b Fz(a)c(>)f FA(0)28 b(and)1154 5006 y Fy(Q)23 b FA(:=)1356 4939 y Fq(\010)1404 5006 y Fz(x)h FA(=)f(\()p Fz(x)1642 5018 y Fv(1)1680 5006 y Fz(;)14 b(:::;)g(x)1870 5018 y Fw(d)1909 5006 y FA(\))23 b Fy(2)g Fx(R)2096 4972 y Fw(d)2192 5006 y FA(:)51 b(0)22 b Fz(<)h(x)2465 5018 y Fw(i)2516 5006 y Fz(<)g(\031)2654 4939 y Fq(\011)2716 5006 y Fz(:)1905 5255 y FA(36)p eop PStoPSsaved restore %%Page: (36,37) 19 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 37 36 bop 515 523 a FA(Assume)27 b(that)h Fz(f)9 b FA(\()p Fz(w)r FA(\))24 b(=)e Fz(O)r FA(\()p Fy(j)p Fz(w)r Fy(j)1492 493 y Fv(3)1531 523 y FA(\).)37 b(Then)28 b(the)g(eigenfunctions)f(of)g (the)h(linearised)e(system)515 623 y(are)34 b(giv)n(en)h(b)n(y)g Fz(')1063 635 y Fw(n)1144 623 y FA(=)h(sin\()p Fz(n)1429 635 y Fv(1)1466 623 y Fz(x)1513 635 y Fv(1)1551 623 y FA(\))14 b(sin\()p Fz(n)1781 635 y Fv(2)1818 623 y Fz(x)1865 635 y Fv(2)1903 623 y FA(\))p Fz(:::)g FA(sin\()p Fz(n)2202 635 y Fw(d)2241 623 y Fz(x)2288 635 y Fw(d)2327 623 y FA(\))36 b(and)f(the)h(corresp)r(onding)e(fre-)515 722 y(quencies)29 b(are)g Fz(!)1040 734 y Fw(n)1112 722 y FA(=)1204 652 y Fq(p)p 1287 652 695 4 v 70 x FA(\()p Fz(n)1369 694 y Fv(2)1369 744 y(1)1425 722 y FA(+)18 b Fz(:::)h FA(+)f Fz(n)1729 694 y Fv(2)1729 747 y Fw(d)1767 722 y FA(\))1799 698 y Fv(2)1855 722 y FA(+)g Fz(a)p FA(,)31 b(where)e Fz(n)e Fy(2)h Fx(Z)2499 692 y Fw(d)2562 722 y FA(and)i Fz(n)2776 734 y Fw(i)2830 722 y Fy(\025)d FA(1.)44 b(T)-7 b(o)29 b(\014t)i(the)515 822 y(ab)r(o)n(v)n(e)25 b(sc)n(heme)i(w)n(e)g(order)e(the)j(basis)e(in)i(suc)n(h)e(a)h(w)n(a)n (y)f(that)h(the)h(frequencies)e(are)g(in)i(non-)515 922 y(decreasing)e(order.)35 b(No)n(w)27 b Fz(X)1424 934 y Fv(0)1484 922 y FA(=)c Fz(L)1629 891 y Fv(2)1665 922 y FA(\()p Fy(Q)p FA(\),)28 b(and)g Fz(X)2079 934 y Fw(s)2137 922 y FA(=)23 b Fz(D)r FA(\(\(\001\001\))2530 891 y Fw(s)2566 922 y FA(\))h Fy(\032)e Fz(H)2785 891 y Fv(4)p Fw(s)2881 922 y FA(endo)n(w)n(ed)27 b(with)515 1021 y(the)c(graph)e(norm.)35 b(If)23 b(the)g(nonlinearit)n(y)f Fz(f)31 b FA(is)23 b(smo)r(oth)f(and)h(o)r(dd)g(\(i.e.)35 b Fz(f)9 b FA(\()p Fy(\000)p Fz(w)r FA(\))24 b(=)e Fy(\000)p Fz(f)9 b FA(\()p Fz(w)r FA(\)\),)515 1121 y(then)25 b(it)g(de\014nes)g(a)g(smo)r(oth)f (map)h(from)f Fz(X)1852 1133 y Fw(s)1913 1121 y FA(to)g(itself)i(for)e (an)n(y)g Fz(s)f(>)f FA([)p Fz(d=)p FA(2])p Fz(=)p FA(4)i(\(see)g (example)515 1220 y(2.5\).)p 3318 1220 4 57 v 3322 1168 50 4 v 3322 1220 V 3372 1220 4 57 v 639 1351 a(In)38 b(the)g(linear)f(appro)n(ximation)f(\()p Fz(g)43 b Fy(\021)c FA(0\))f(the)g(general)e(solution)h(of)h(\(10.1\))f(is)h(the)515 1451 y(sup)r(erp)r(osition)18 b(of)i(the)f(linear)g(normal)f(mo)r(des,) j(i.e.)34 b(of)19 b(the)g(families)h(of)f(p)r(erio)r(dic)g(solutions) 1248 1627 y Fz(x)1295 1593 y Fv(\()p Fw(j)s Fv(\))1382 1627 y FA(\()p Fz(t)p FA(\))24 b(=)e(\()p Fz(a)1663 1639 y Fw(j)1712 1627 y FA(cos)o(\()p Fz(!)1907 1639 y Fw(j)1942 1627 y Fz(t)p FA(\))d(+)f Fz(b)2142 1639 y Fw(j)2191 1627 y FA(sin)o(\()p Fz(!)2376 1639 y Fw(j)2411 1627 y Fz(t)p FA(\)\))p Fz(')2559 1639 y Fw(j)2623 1627 y Fz(:)521 b FA(\(10.6\))515 1804 y(Fix)38 b(one)g(of)g(the)h(families,)i (sa)n(y)c Fz(x)1636 1774 y Fv(\(1\))1726 1804 y FA(.)69 b(T)-7 b(o)38 b(ensure)f(its)i(p)r(ersistence)f(in)g(the)h(nonlinear) 515 1904 y(problem)27 b(w)n(e)g(mak)n(e)g(the)h(follo)n(wing)e (assumptions:)545 2081 y(H1\))41 b(\(Nonresonance\))22 b(F)-7 b(or)22 b(small)g(enough)g Fz(\015)28 b(>)23 b FA(0)f(there)h(exists)f(a)g(closed)g(set)h Fz(W)3110 2093 y Fw(\015)3176 2081 y Fy(\032)g Fx(R)3318 2051 y Fv(+)722 2180 y FA(ha)n(ving)g Fz(!)1038 2192 y Fv(1)1099 2180 y FA(as)g(an)h(accum)n(ulation)f(p)r(oin)n(t)h(b)r(oth)g(from)f (the)i(righ)n(t)e(and)h(from)f(the)h(left,)722 2280 y(and)k(suc)n(h)f (that)h(for)f(an)n(y)g Fz(!)e Fy(2)f Fz(W)1769 2292 y Fw(\015)1840 2280 y FA(one)j(has)1372 2481 y Fy(j)p Fz(!)s(l)20 b Fy(\000)e Fz(!)1630 2493 y Fw(j)1664 2481 y Fy(j)24 b(\025)1808 2425 y Fz(\015)p 1808 2462 48 4 v 1819 2538 a(l)1893 2481 y(;)97 b Fy(8)14 b Fz(l)24 b Fy(\025)e FA(1)28 b Fz(;)41 b Fy(8)p Fz(j)27 b Fy(\025)c FA(2)p 2669 2481 4 57 v 2673 2429 50 4 v 2673 2481 V 2722 2481 4 57 v 585 w(\(10.7\))545 2718 y(H2\))41 b(\(Nondegeneracy\))21 b(Let)i Fz(g)1542 2730 y Fw(r)1578 2718 y FA(\()p Fz(x)p FA(\))h(b)r(e)f(the)g(\014rst)f(non-v)-5 b(anishing)21 b(\(homogeneous\))g(T)-7 b(a)n(y-)722 2818 y(lor)27 b(p)r(olynomial)g (of)h Fz(g)s FA(.)36 b(Assume)28 b(that)g Fz(r)d Fy(\025)e FA(3)k(and)h Fz(\014)2389 2830 y Fv(0)2449 2818 y Fy(6)p FA(=)23 b(0,)k(where)1225 3040 y Fz(\014)1272 3052 y Fv(0)1332 3040 y FA(:=)1443 2923 y Fq(\032)1547 2989 y Fy(h)p Fz(g)1619 3001 y Fw(r)1656 2989 y FA(\()p Fz(')1742 3001 y Fv(1)1780 2989 y FA(\))p Fz(;)14 b(')1903 3001 y Fv(1)1940 2989 y Fy(i)1972 3001 y Fv(0)2135 2989 y FA(if)90 b Fz(r)c FA(is)169 b(o)r(dd)p Fz(;)1505 3089 y Fy(h)p Fz(g)1577 3101 y Fw(r)r Fv(+1)1698 3089 y FA(\()p Fz(')1784 3101 y Fv(1)1822 3089 y FA(\))p Fz(;)14 b(')1945 3101 y Fv(1)1982 3089 y Fy(i)2014 3101 y Fv(0)2135 3089 y FA(if)90 b Fz(r)c FA(is)c(ev)n(en)p Fz(:)p 2806 3089 V 2810 3036 50 4 v 2810 3089 V 2859 3089 4 57 v 3167 3040 a FA(\(10.8\))639 3346 y(Denoting)28 b Fz(\030)1034 3358 y Fv(1)1071 3346 y FA(\()p Fz(!)1155 3358 y Fv(1)1193 3346 y Fz(t)p FA(\))23 b(=)g(cos)o(\()p Fz(!)1561 3358 y Fv(1)1598 3346 y Fz(t)p FA(\))p Fz(')1714 3358 y Fv(1)1779 3346 y FA(one)28 b(has)515 3508 y FB(Theorem)i(10.3.)39 b Fp(Supp)l(ose)30 b(the)g(assumptions)g(H1,H2)g(hold.)40 b(Then)31 b(ther)l(e)e(exists)g(a)h(set)515 3607 y Fy(E)45 b(\032)37 b Fx(R)43 b Fp(having)c(zer)l(o)f(as)h(an)e(ac)l(cumulation)h (p)l(oint,)j(a)d(p)l(ositive)h Fz(!)2698 3619 y Fu(\003)2736 3607 y Fp(,)h(and)e(a)h(family)g(of)515 3707 y(p)l(erio)l(dic)32 b(solutions)e Fy(f)p Fz(x)1263 3719 y Fw(\017)1294 3707 y FA(\()p Fz(t)p FA(\))p Fy(g)1430 3719 y Fw(\017)p Fu(2E)1577 3707 y Fp(of)h(\(10.1\))g(with)g(fr)l(e)l(quencies)f Fy(f)p Fz(!)2626 3677 y Fw(\017)2657 3707 y Fy(g)2699 3719 y Fw(\017)p Fu(2E)2845 3707 y Fp(ful\014l)t(ling)984 3884 y FA(sup)1034 3950 y Fw(t)1123 3884 y Fy(k)p Fz(x)1212 3896 y Fw(\017)1243 3884 y FA(\()p Fz(t)p FA(\))19 b Fy(\000)f Fz(\017\030)1509 3896 y Fv(1)1547 3884 y FA(\()p Fz(t!)1664 3849 y Fw(\017)1696 3884 y FA(\))p Fy(k)1770 3896 y Fw(s)1828 3884 y Fy(\024)k Fz(C)6 b(\017)2014 3849 y Fw(r)2081 3884 y Fz(;)99 b Fy(j)p Fz(!)2281 3849 y Fw(\017)2331 3884 y Fy(\000)18 b Fz(!)2466 3896 y Fv(1)2503 3884 y Fy(j)23 b(\024)f Fz(C)6 b(\017)2735 3849 y Fw(r)r Fu(\000)p Fv(1)2887 3884 y Fz(:)257 b FA(\(10.9\))515 4106 y Fp(Mor)l(e)l(over,)26 b(the)d(set)f Fy(E)30 b Fp(is)24 b(in)e(one)i(to)e(one)h(c)l(orr)l(esp)l(ondenc)l(e)h(either)g (with)f Fz(W)2842 4118 y Fw(\015)2889 4106 y Fy(\\)s FA([)p Fz(!)3022 4118 y Fv(1)3060 4106 y Fz(;)14 b(!)3149 4118 y Fv(1)3189 4106 y FA(+)s Fz(!)3309 4118 y Fu(\003)3347 4106 y FA(\))515 4205 y Fp(if)30 b Fz(\014)642 4217 y Fv(0)703 4205 y Fz(<)22 b FA(0)p Fp(,)30 b(or)g(with)g Fz(W)1252 4217 y Fw(\015)1314 4205 y Fy(\\)19 b FA(\()p Fz(!)1472 4217 y Fv(1)1527 4205 y Fy(\000)f Fz(!)1662 4217 y Fu(\003)1700 4205 y Fz(;)c(!)1789 4217 y Fv(1)1826 4205 y FA(])30 b Fp(if)g Fz(\014)2006 4217 y Fv(0)2067 4205 y Fz(>)22 b FA(0)p Fp(.)515 4367 y FB(Pro)s(of.)33 b FA(W)-7 b(e)21 b(consider)e(only)g(the)i(case)e(of)h(o)r(dd)g Fz(r)r FA(,)i(the)f(general)d(case)h(can)h(b)r(e)g(obtained)g(b)n(y)f (a)515 4466 y(sligh)n(tly)h(di\013eren)n(t)g(treatmen)n(t)g(of)h(the)f (forthcoming)g(equation)g Fz(!)s FA(.)34 b(W)-7 b(e)21 b(are)e(lo)r(oking)g(for)h(an)515 4566 y Fz(X)584 4578 y Fw(s)619 4566 y FA(-v)-5 b(alued)28 b(function)g Fz(q)s FA(\()p Fz(t)p FA(\))h(whic)n(h)f(is)g(2)p Fz(\031)s FA(-p)r(erio)r(dic)f(and)h(rev)n(ersible)e(\(i.e.,)j Fz(q)s FA(\()p Fz(t)p FA(\))24 b(=)f Fz(q)s FA(\()p Fy(\000)p Fz(t)p FA(\)\),)515 4666 y(and)30 b(for)g(a)f(p)r(ositiv)n(e)h Fz(!)s FA(,)h(close)f(to)g Fz(!)1657 4678 y Fv(1)1694 4666 y FA(,)h(suc)n(h)f(that)g Fz(q)s FA(\()p Fz(!)s(t)p FA(\))h(is)f(a)g(solution)g(of)g(\(10.1\).)44 b(They)515 4765 y(m)n(ust)27 b(satisfy)h(the)g(equation)1334 4991 y Fz(L)1391 5003 y Fw(!)1439 4991 y Fz(q)e FA(=)d Fz(g)s FA(\()p Fz(q)s FA(\))k Fz(;)97 b(L)1941 5003 y Fw(!)2012 4991 y FA(:=)23 b Fz(!)2178 4957 y Fv(2)2240 4935 y Fz(d)2283 4904 y Fv(2)p 2225 4972 111 4 v 2225 5048 a Fz(dt)2298 5024 y Fv(2)2363 4991 y FA(+)18 b Fz(A)28 b(;)566 b FA(\(10.10\))1905 5255 y(37)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 38 37 bop 515 523 a FA(whic)n(h)35 b(will)h(b)r(e)g(considered)e(as)h (an)g Fz(!)s FA(-dep)r(enden)n(t)h(functional)f(equation)g(in)h(the)g (space)515 623 y Fy(H)27 b(\032)g Fz(H)780 593 y Fv(1)817 623 y FA(\()p Fx(T)p Fz(;)14 b(X)1011 635 y Fw(s)1045 623 y FA(\),)31 b(formed)f(b)n(y)f(the)h(rev)n(ersible)e(p)r(erio)r (dic)i(functions.)44 b(Equation)28 b(\(10.10\))515 722 y(is)23 b(studied)h(using)g(the)g(Ly)n(apuno)n(v{Sc)n(hmidt)e(decomp)r (osition,)h(namely)h(b)n(y)f(decomp)r(osing)515 822 y(it)32 b(in)n(to)g(an)g(equation)g(on)g(Ker)p Fz(L)1551 834 y Fw(!)1593 842 y Ft(1)1658 822 y Fy(\021)14 b FA(span\()p Fz(\030)1972 834 y Fv(1)2009 822 y FA(\))32 b(and)g(an)g(equation)g(on) g(its)g(complemen)n(t)515 922 y Fz(R)q FA(.)k(Precisely)-7 b(,)24 b(denote)i(b)n(y)f Fz(Q)h FA(the)g(pro)5 b(jector)24 b(on)h Fz(\030)2119 934 y Fv(1)2182 922 y FA(and)h(b)n(y)f Fz(P)38 b FA(the)26 b(pro)5 b(jector)24 b(on)h Fz(R)h FA(and)515 1021 y(mak)n(e)g(the)h(Ansatz)g Fz(q)f FA(=)d Fz(\017\030)1373 1033 y Fv(1)1427 1021 y FA(+)17 b Fz(\017)1543 991 y Fw(r)1580 1021 y Fz(u)p FA(,)26 b(where)h Fz(u)22 b Fy(2)i Fz(R)q FA(.)36 b(Then)27 b(\(10.10\))f(is)h(equiv)-5 b(alen)n(t)27 b(to)g(the)515 1121 y(system)1645 1303 y Fz(!)1700 1269 y Fv(2)1760 1303 y FA(=)c Fz(!)1903 1269 y Fv(2)1900 1323 y(1)1958 1303 y FA(+)18 b Fz(\014)t(\017)2126 1269 y Fw(r)r Fu(\000)p Fv(1)3125 1303 y FA(\(10.11\))1469 1427 y Fz(L)1526 1439 y Fw(!)1573 1427 y Fz(u)23 b FA(=)f Fz(P)12 b(g)1836 1439 y Fw(r)1873 1427 y FA(\()p Fz(\030)1941 1439 y Fv(1)1979 1427 y FA(\))18 b(+)g Fz(P)12 b(G)p FA(\()p Fz(\017;)i(u)p FA(\))700 b(\(10.12\))1449 1552 y Fy(\000)p Fz(\014)t(\030)1601 1564 y Fv(1)1662 1552 y FA(=)22 b Fz(Qg)1855 1564 y Fw(r)1891 1552 y FA(\()p Fz(\030)1959 1564 y Fv(1)1997 1552 y FA(\))d(+)f Fz(QG)p FA(\()p Fz(\017;)c(u)p FA(\))680 b(\(10.13\))515 1734 y(for)36 b(the)h(unkno)n(wns)f(\()p Fz(\017;)14 b(u;)g(\014)t FA(\).)65 b(Here)36 b Fz(G)h FA(con)n(tains)f(all)h(higher)f(order)f (corrections)g(and)515 1834 y Fz(!)d Fy(2)d Fz(W)761 1846 y Fw(\015)835 1834 y FA(is)i(a)g(parameter.)47 b(The)31 b(equations)f(\(10.11\),)h(\(10.12\))f(and)h(\(10.13\))f(are)g(called) 515 1933 y(the)e Fz(!)s FA(,)f(the)h(P)f(and)h(the)g(Q)f(equation,)g (resp)r(ectiv)n(ely)-7 b(.)639 2033 y(First)32 b(one)g(solv)n(es)f(the) h(P)g(equation)g(\(10.12\).)49 b(T)-7 b(o)32 b(this)g(end)h(one)f(has)f (to)h(in)n(v)n(ert)g(the)515 2132 y(linear)g(op)r(erator)g Fz(L)1149 2144 y Fw(!)1197 2062 y Fq(\014)1197 2112 y(\014)1224 2166 y Fw(R)1279 2132 y FA(.)54 b(Its)34 b(eigenfunctions)f(are)f (cos\()p Fz(l)r(t)p FA(\))p Fz(')2464 2144 y Fw(j)2499 2132 y FA(,)j(and)e(the)h(corresp)r(onding)515 2232 y(eigen)n(v)-5 b(alues)26 b(are)955 2414 y Fz(\025)1003 2426 y Fw(j)s(l)1082 2414 y FA(=)d Fy(\000)p Fz(l)1262 2380 y Fv(2)1298 2414 y Fz(!)1353 2380 y Fv(2)1409 2414 y FA(+)18 b Fz(!)1547 2380 y Fv(2)1544 2435 y Fw(j)1607 2414 y FA(=)k(\()p Fz(l)r(!)f FA(+)d Fz(!)1961 2426 y Fw(j)1996 2414 y FA(\)\()p Fz(!)2112 2426 y Fw(j)2165 2414 y Fy(\000)g Fz(l)r(!)s FA(\))27 b Fz(;)42 b(j)28 b Fy(\025)23 b FA(2)k Fz(;)41 b(l)25 b Fy(\025)e FA(1)p Fz(:)515 2607 y FA(By)39 b(\(10.7\),)j Fy(j)p Fz(\025)1006 2619 y Fw(j)s(l)1063 2607 y Fy(j)i Fz(>)e(C)6 b(\015)f FA(.)74 b(So)39 b(\()p Fz(L)1663 2619 y Fw(!)1711 2537 y Fq(\014)1711 2586 y(\014)1739 2640 y Fw(R)1793 2607 y FA(\))1825 2577 y Fu(\000)p Fv(1)1954 2607 y FA(exists)h(and)f(is)h(b)r(ounded.)74 b(Applying)40 b(this)515 2707 y(op)r(erator)23 b(to)i(the)h(P)f(equation)g(and)g (using)f(the)i(implicit)g(function)g(theorem)f(one)g(obtains)515 2806 y(a)j(smo)r(oth)h(function)g Fz(u)p FA(\()p Fz(\017)p FA(\))g(that)g(dep)r(ends)g(parametrically)e(on)i Fz(!)e Fy(2)f Fz(W)2792 2818 y Fw(\015)2864 2806 y FA(and)i(solv)n(es)g(the) 515 2906 y(P)f(equation.)639 3006 y(Inserting)18 b Fz(u)p FA(\()p Fz(\017)p FA(\))h(in)g(the)g Fz(Q)g FA(equation)f(one)g (determines)h(the)g(parameter)e Fz(\014)23 b FA(as)18 b(a)g(function)515 3105 y(of)28 b Fz(\017)p FA(.)40 b(In)29 b(particular)e(one)h(has)g Fz(\014)t FA(\()p Fz(\017)p FA(\))e(=)e Fz(C)6 b(\014)1874 3117 y Fv(0)1911 3105 y FA(+higher)28 b(order)f(corrections,)g(where)h Fz(C)j(>)24 b FA(0.)515 3205 y(Inserting)31 b Fz(\014)t FA(\()p Fz(\017)p FA(\))h(in)g(the)g Fz(!)i FA(equation)d(one)g(gets)g(an)h(equation)f (for)g Fz(\017)g FA(\(remem)n(b)r(er)g(that)h Fz(!)515 3304 y FA(is)f(\014xed\),)h(whic)n(h)f(is)h(a)e(p)r(erturbation)h(of)g (the)h(equation)e Fz(!)2396 3274 y Fv(2)2454 3304 y Fy(\000)20 b Fz(!)2594 3274 y Fv(2)2591 3325 y(1)2660 3304 y FA(=)29 b Fz(C)6 b(\014)2866 3316 y Fv(0)2904 3304 y Fz(\017)2938 3274 y Fw(r)r Fu(\000)p Fv(1)3059 3304 y FA(.)48 b(By)31 b(the)515 3404 y(nondegeneracy)d(this)j(can)f(b)r(e)h(reduced)f(to)g(a) g(\014xed)h(p)r(oin)n(t)f(equation)g(for)g Fz(\017)2931 3374 y Fw(r)r Fu(\000)p Fv(1)3083 3404 y FA(whic)n(h)g(is)515 3504 y(solv)-5 b(able)27 b(b)n(y)g(the)h(con)n(traction)e(mapping)h (principle.)p 3318 3504 4 57 v 3322 3451 50 4 v 3322 3504 V 3372 3504 4 57 v 515 3636 a Fp(R)l(emark)44 b FA(10.4)p Fp(.)f FA(The)34 b(theorem)e(holds)h(also)f(in)i(the)f(case)f Fz(r)k FA(=)c(2,)i(but)f(in)h(this)f(case)g(the)515 3736 y(nondegeneracy)25 b(condition)j(tak)n(es)e(a)i(more)e(complicated)i (form.)p 3318 3736 V 3322 3683 50 4 v 3322 3736 V 3372 3736 4 57 v 639 3869 a(Theorem)39 b(10.3)g(w)n(as)g(pro)n(v)n(ed)g(in)h ([Bam00)n(].)75 b(The)40 b(tec)n(hnique)g(of)g(the)g(Ly)n(apuno)n(v{) 515 3968 y(Sc)n(hmidt)26 b(decomp)r(osition)g(w)n(as)f(used)g(for)h (the)g(\014rst)g(time)g(to)g(construct)g(families)g(of)f(p)r(eri-)515 4068 y(o)r(dic)h(solutions)g(in)g(PDEs)g(b)n(y)g(Craig)f(and)h(W)-7 b(a)n(yne)26 b([CW93)o(])h(who)f(considered)f(the)i(mo)r(del)515 4167 y(problem)i(of)g(the)h(w)n(a)n(v)n(e)e(equation)h(with)h(p)r(erio) r(dic)f(b)r(oundary)g(conditions)g(\(see)g(example)515 4267 y(10.1\);)d(w)n(e)i(will)f(rep)r(ort)g(on)h(this)f(w)n(ork)g(in)g (Section)h(10.4.)515 4400 y Fp(Example)43 b FA(10.5)p Fp(.)h FA(Consider)34 b(the)g(nonlinear)g(w)n(a)n(v)n(e)e(equation)i (with)h(p)r(erio)r(dic)f(b)r(oundary)515 4499 y(conditions)g(\(see)h (example)f(10.1\).)57 b(Let)35 b Fz(!)1894 4511 y Fv(1)1966 4499 y FA(b)r(e)g(suc)n(h)g(that)g Fz(!)2520 4511 y Fv(1)2592 4499 y Fy(6)p FA(=)f Fz(!)2743 4511 y Fw(j)2813 4499 y FA(for)g(eac)n(h)g Fz(j)40 b Fy(6)p FA(=)34 b(1.)515 4608 y(Decomp)r(ose)22 b Fz(V)42 b FA(in)n(to)22 b(its)h(a)n(v)n(erage) d Fz(a)j FA(and)f(a)g(part)2077 4587 y(~)2065 4608 y Fz(V)41 b FA(of)23 b(zero)f(a)n(v)n(erage,)f(then)i(condition)f(H1)515 4707 y(is)i(satis\014ed)f(if)h Fz(a)g FA(b)r(elongs)f(to)h(an)g(uncoun) n(table)f(set)h(whic)n(h)g(is)g(dense)f(in)i(a)e(neigh)n(b)r(ourho)r(o) r(d)515 4807 y(of)35 b(the)g(origin)f(\(for)h(the)h(pro)r(of)e(see)h (Lemma)g(3.1)f(of)h([BP02)n(]\).)60 b(Condition)35 b(H2)g(can)g(b)r(e) 515 4907 y(expressed)29 b(in)h(terms)g(of)g(the)g(eigenfunctions)g(of)g (the)h(Sturm{Liouville)e(op)r(erator.)43 b(If)31 b(it)515 5006 y(holds,)d(then)i(Theorem)e(10.3)f(applies)h(and)h(ensures)f(p)r (ersistence)g(of)h(the)g(corresp)r(onding)1905 5255 y(38)p eop PStoPSsaved restore %%Page: (38,39) 20 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 39 38 bop 515 523 a FA(family)32 b(of)g(p)r(erio)r(dic)g(orbits.)51 b(Note)32 b(that,)i(in)e(a)g(di\013erence)g(with)h(the)f(case)g(of)g (Diric)n(hlet)515 623 y(b)r(oundary)c(conditions)g(\(see)h(example)f (5.3\),)h(the)g(nonlinearit)n(y)f(do)r(es)h(not)g(need)g(to)f(ha)n(v)n (e)515 722 y(some)f(particular)f(parit)n(y)-7 b(.)p 3318 722 4 57 v 3322 670 50 4 v 3322 722 V 3372 722 4 57 v 515 853 a Fp(Example)33 b FA(10.6)p Fp(.)j FA(Consider)22 b(the)i(nonlinear)e(plate)h(equation)g(\(see)g(example)g(10.2\).)34 b(In)24 b(the)515 953 y(case)c Fz(d)j FA(=)g(1)e(all)g(the)g (frequencies)g(are)f(simple)h(and)g(the)h(assumption)f(H1)g(is)g (satis\014ed)f(if)i Fz(a)f FA(is)515 1053 y(c)n(hosen)i(in)h(a)g (subset)g(of)g Fx(R)1332 1023 y Fv(+)1417 1053 y FA(ha)n(ving)f(full)i (measure.)34 b(In)25 b(the)f(case)f Fz(d)g(>)g FA(1,)h(all)g(the)h (frequen-)515 1152 y(cies)h(are)g(m)n(ultiple)h(except)f(the)h (smallest)g(one.)36 b(T)-7 b(aking)26 b(for)g Fz(!)2491 1164 y Fv(1)2554 1152 y FA(the)h(smallest)f(frequency)-7 b(,)515 1252 y(H1)32 b(is)g(ful\014lled)h(if)g Fz(a)f FA(b)r(elongs)g(to)g(a)g(dense)g(uncoun)n(table)g(subset)h(of)f ([0,1/4].)49 b(H2)32 b(holds)515 1352 y(trivially)27 b(pro)n(vided)g(the)i(T)-7 b(a)n(ylor)26 b(expansion)h(of)h Fz(f)37 b FA(at)28 b(zero)f(do)r(es)g(not)h(v)-5 b(anish)28 b(iden)n(tically)515 1451 y(\(remem)n(b)r(er)c(that)h Fz(f)9 b FA(\()p Fy(\000)p Fz(w)r FA(\))24 b(=)e Fz(f)9 b FA(\()p Fz(w)r FA(\)\).)38 b(Then)25 b(Theorem)f(10.3)f(ensures)h(p)r (ersistence)h(of)g(the)515 1551 y(corresp)r(onding)g(family)j(of)g(p)r (erio)r(dic)f(orbits)g(\(for)g(details)h(see)f([BP02)n(]\).)p 3318 1551 V 3322 1498 50 4 v 3322 1551 V 3372 1551 4 57 v 515 1783 a Fs(10.3)112 b(The)38 b(resonan)m(t)g(case)515 1936 y FA(It)29 b(is)g(p)r(ossible)g(to)g(generalise)e(the)j(ab)r(o)n (v)n(e)e(theorem)g(to)h(the)h(case)e(when)h(the)h(frequencies)515 2035 y(satisfy)35 b(some)g(resonance)e(relations.)59 b(W)-7 b(e)36 b(will)g(consider)e(only)h(the)h(Lagrangian)d(case,)515 2135 y(when)28 b Fz(g)d FA(=)e Fy(\000r)p Fz(H)7 b FA(.)639 2235 y(Fix)28 b(a)g(frequency)f Fz(!)1286 2247 y Fv(1)1351 2235 y FA(of)h(the)g(linearised)f(system.)37 b(W)-7 b(e)29 b(replace)e(the)h(assumption)f(H1)515 2334 y(b)n(y)g(the)h(follo)n (wing)f(one:)484 2497 y(H1R\))41 b(F)-7 b(or)23 b(an)n(y)f(small)h (enough)g Fz(\015)28 b FA(there)23 b(exists)g(a)g(closed)f(set)h Fz(W)2526 2509 y Fw(\015)2592 2497 y Fy(\032)g Fx(R)2734 2467 y Fv(+)2818 2497 y FA(ha)n(ving)g Fz(!)3134 2509 y Fv(1)3194 2497 y FA(as)f(an)722 2597 y(accum)n(ulation)29 b(p)r(oin)n(t)h(b)r(oth)g(from)f(the)h(righ)n(t)f(and)g(from)g(the)h (left,)h(and)f(suc)n(h)f(that)722 2696 y(for)e(an)n(y)g Fz(!)f Fy(2)d Fz(W)1240 2708 y Fw(\015)1311 2696 y FA(one)k(has)1015 2899 y(either)166 b Fy(j)p Fz(!)s(l)20 b Fy(\000)e Fz(!)1647 2911 y Fw(j)1681 2899 y Fy(j)24 b(\025)1825 2843 y Fz(\015)p 1825 2880 48 4 v 1836 2956 a(l)1910 2899 y(;)180 b FA(or)165 b Fz(l)r(!)2432 2911 y Fv(1)2487 2899 y Fy(\000)18 b Fz(!)2622 2911 y Fw(j)2680 2899 y FA(=)23 b(0)p Fz(:)292 b FA(\(10.14\))p 3318 3099 4 57 v 3322 3046 50 4 v 3322 3099 V 3372 3099 4 57 v 639 3262 a(T)-7 b(o)27 b(pass)g(to)h(the)g (nondegeneracy)d(assumption,)j(w)n(e)f(de\014ne)h(the)g(resonan)n(t)e (set)h(as)1204 3440 y Fy(I)1249 3452 y Fw(R)1326 3440 y FA(:=)c Fy(f)o Fz(k)j Fy(\025)d FA(1)50 b(:)h Fy(9)p Fz(l)25 b Fy(\025)d FA(1)50 b(:)h Fz(l)r(!)2229 3452 y Fv(1)2284 3440 y Fy(\000)18 b Fz(!)2419 3452 y Fw(k)2483 3440 y FA(=)k(0)p Fy(g)13 b Fz(;)435 b FA(\(10.15\))515 3619 y(and)33 b(denote)h(b)n(y)g Fy(N)46 b FA(the)34 b(closure)f(in)h(the)g(graph)f(norm)g(of)h Fz(D)r FA(\()p Fz(A)p FA(\))h(of)e(the)i(linear)e(space,)515 3718 y(generated)f(b)n(y) h Fy(f)p Fz(')1116 3730 y Fw(k)1157 3718 y Fy(g)1199 3730 y Fw(k)q Fu(2I)1318 3738 y Fn(R)1368 3718 y FA(.)55 b(Note)33 b(that)h(all)f(solutions)g(of)g(the)h(linearised)e(system)i (with)515 3818 y(initial)c(datum)g(in)g Fy(N)42 b FA(and)30 b(v)-5 b(anishing)29 b(initial)h(v)n(elo)r(cit)n(y)f(are)g(p)r(erio)r (dic)g(of)h(p)r(erio)r(d)g(2)p Fz(\031)s(=!)3320 3830 y Fv(1)3356 3818 y FA(.)515 3918 y(Let)c Fz(H)731 3930 y Fw(r)793 3918 y FA(b)r(e)h(the)f(\014rst)g(non)f(v)-5 b(anishing)26 b(T)-7 b(a)n(ylor)24 b(co)r(e\016cien)n(t)h(of)h Fz(H)33 b FA(and,)26 b(for)f Fz(x)f Fy(2)f(N)12 b FA(,)27 b(de\014ne)515 4017 y(the)h(a)n(v)n(erage)c(of)k Fz(H)1119 4029 y Fw(r)1183 4017 y FA(b)n(y)1230 4257 y Fy(h)p Fz(H)1331 4269 y Fw(r)1368 4257 y Fy(i)p FA(\()p Fz(x)p FA(\))c(:=)1657 4200 y Fz(!)1709 4212 y Fv(1)p 1656 4237 92 4 v 1656 4314 a FA(2)p Fz(\031)1771 4144 y Fq(Z)1854 4164 y Fv(2)p Fw(\031)r(=!)2004 4172 y Ft(1)1817 4332 y Fv(0)2054 4257 y Fz(H)2123 4269 y Fw(r)2160 4257 y FA(\(cos\()p Fz(At)p FA(\))p Fz(x)p FA(\))p Fz(dt)29 b(:)515 4482 y FA(Consider)d(the)i(h)n (yp)r(ersurface)f Fy(S)i(\032)23 b(N)40 b FA(of)27 b(the)h(p)r(oin)n (ts)g Fz(x)23 b Fy(2)h(N)40 b FA(suc)n(h)27 b(that)h Fy(h)p Fz(x)p FA(;)14 b Fz(Ax)p Fy(i)3133 4494 y Fv(0)3195 4482 y FA(=)23 b(1.)484 4644 y(H2R\))41 b(There)34 b(exists)g(a)f (nondegenerate)g(critical)h(p)r(oin)n(t)g Fz(x)2390 4656 y Fv(0)2462 4644 y FA(of)g(the)g(functional)h Fy(h)p Fz(H)3210 4656 y Fw(r)3247 4644 y Fy(i)3279 4574 y Fq(\014)3279 4624 y(\014)3307 4677 y Fu(S)3356 4644 y FA(,)722 4744 y(and)28 b(the)g(corresp)r(onding)d(Lagrange)h(m)n(ultiplier)h(do)r(es) h(not)f(v)-5 b(anish.)p 3318 4744 4 57 v 3322 4691 50 4 v 3322 4744 V 3372 4744 4 57 v 639 4907 a(Denote)33 b(b)n(y)f Fz(\030)1085 4919 y Fv(0)1122 4907 y FA(\()p Fz(!)1206 4919 y Fv(1)1243 4907 y Fz(t)p FA(\))h(the)g(solution)e(of)i (the)f(linearised)g(system)g(with)g(initial)h(datum)515 5006 y Fz(x)562 5018 y Fv(0)627 5006 y FA(and)28 b(v)-5 b(anishing)27 b(initial)h(v)n(elo)r(cit)n(y)-7 b(.)1905 5255 y(39)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 40 39 bop 515 523 a FB(Theorem)30 b(10.7.)39 b Fp([BP01)r(])30 b(Supp)l(ose)f(the)h(assumptions)f(H1R,)h(H2R)f(hold.)40 b(Then)30 b(ther)l(e)515 623 y(exists)37 b(a)h(family)i(of)f(p)l(erio)l (dic)g(solutions)f Fy(f)p Fz(x)1960 635 y Fw(\017)1992 623 y FA(\()p Fz(t)p FA(\))p Fy(g)2128 635 y Fw(\017)p Fu(2E)2283 623 y Fp(of)h(\(10.1\))g(with)g(fr)l(e)l(quencies)f Fz(!)3322 593 y Fw(\017)3354 623 y Fp(,)515 722 y(satisfying)857 905 y FA(sup)907 971 y Fw(t)996 905 y Fy(k)p Fz(x)1085 917 y Fw(\017)1117 905 y FA(\()p Fz(t)p FA(\))19 b Fy(\000)f Fz(\017\030)1383 917 y Fv(0)1420 905 y FA(\()p Fz(t!)1537 871 y Fw(\017)1569 905 y FA(\))p Fy(k)1643 917 y Fw(s)1701 905 y Fy(\024)23 b Fz(C)6 b(\017)1888 871 y Fw(r)1954 905 y Fz(;)99 b Fy(j)p Fz(!)2154 871 y Fw(\017)2204 905 y Fy(\000)18 b Fz(!)2339 917 y Fv(1)2376 905 y Fy(j)23 b(\024)g Fz(C)6 b(\017)2609 871 y Fw(r)r Fu(\000)p Fv(1)2760 905 y Fz(:)342 b FA(\(10.16\))515 1128 y Fp(The)30 b(set)g Fy(E)37 b Fp(has)30 b(the)g(same)g(pr)l(op)l(erties)h(as)f(in)g(the)g (nonr)l(esonant)f(c)l(ase.)639 1294 y FA(The)j(pro)r(of)e(is)h (obtained)g(b)n(y)g(pro)r(ceeding)f(as)h(in)g(the)h(nonresonan)n(t)d (case.)47 b(The)31 b(only)515 1394 y(di\013erence)24 b(is)f(that)i(in)f(this)g(case)f(the)h(k)n(ernel)f(of)h Fz(L)2090 1406 y Fw(!)2132 1414 y Ft(1)2192 1394 y FA(is)g(no)f(longer) g(one)g(dimensional,)h(but)515 1493 y(is)d(isomorphic)f(to)h Fy(N)33 b FA(\(the)22 b(isomorphism)e(b)r(eing)h(giv)n(en)g(b)n(y)g (the)g(map)g Fz(x)j Fy(7!)f FA(cos)o(\()p Fz(At=!)3207 1505 y Fv(1)3244 1493 y FA(\))p Fz(x)p FA(\).)515 1593 y(So)k(the)h(Q)f(equation)g(can)g(b)r(e)h(transformed)e(in)n(to)h(an)h (equation)f(in)g Fy(N)12 b FA(.)38 b(The)27 b(latter)g(turns)515 1693 y(out)k(to)f(b)r(e)i(a)e(p)r(erturbation)g(of)h(the)g(equation)g (for)f(the)h(critical)g(p)r(oin)n(ts)f(of)h Fy(h)p Fz(H)3044 1705 y Fw(r)3081 1693 y Fy(i)3113 1622 y Fq(\014)3113 1672 y(\014)3141 1726 y Fu(S)3190 1693 y FA(,)h(and)515 1792 y(the)38 b(nondegeneracy)d(condition)i(H2R)h(allo)n(ws)e(to)h (solv)n(e)f(it)i(b)n(y)f(the)h(implicit)g(function)515 1892 y(theorem.)639 1992 y(Applying)29 b(the)g(ab)r(o)n(v)n(e)e (theorem,)h(one)g(can)g(construct)f(coun)n(tably)h(man)n(y)g(families)g (of)515 2091 y(p)r(erio)r(dic)f(solutions)g(of)h(the)g Fz(\036)1467 2061 y Fv(4)1504 2091 y FA(-mo)r(del)1245 2274 y Fz(w)1304 2286 y Fw(tt)1377 2274 y Fy(\000)18 b Fz(w)1519 2286 y Fw(xx)1622 2274 y FA(=)k Fy(\006)p Fz(w)1835 2240 y Fv(3)1891 2274 y FA(+)c(higher)27 b(order)g(terms)515 2457 y(with)21 b(Diric)n(hlet)h(b)r(oundary)e(conditions,)i(and)f(also) f(higher)g(frequency)h(p)r(erio)r(dic)g(solutions)515 2556 y(of)38 b(the)i(nonlinear)d(plate)i(equation)f(of)h(example)f (10.2)g(\(see)g([BP01)o(,)h(BP02)n(],)j(see)c(also)515 2656 y([LS88)o(,)28 b(Bou99b)n(]\).)639 2755 y(In)h(general)d(it)j(is)f (di\016cult)h(to)f(c)n(hec)n(k)f(condition)h(H2R.)g(In)g(the)h(case)e (of)h(Hamiltonian)515 2855 y(systems)i(with)g Fz(n)e(<)f Fy(1)j FA(degrees)f(of)i(freedom,)f(top)r(ological)f(argumen)n(ts)g (allo)n(w)g(to)i(a)n(v)n(oid)515 2955 y(it.)70 b(Indeed,)42 b(the)c(W)-7 b(einstein{Moser)38 b(theorem)g(\(see)h([W)-7 b(ei73)o(,)39 b(Mos76)n(]\))g(ensures)f(that)515 3054 y(close)33 b(to)h(a)f(minim)n(um)i(of)f(the)h(energy)d(on)i(eac)n(h)f (surface)h(of)g(a)f(constan)n(t)g(energy)g(there)515 3154 y(exist)c(at)g(least)f Fz(n)h FA(p)r(erio)r(dic)g(orbit.)41 b(In)29 b(general)f(they)h(do)g(not)g(form)f(regular)g(families.)41 b(A)515 3254 y(corresp)r(onding)25 b(result)h(for)g(PDEs)g(is)h(not)g (a)n(v)-5 b(ailable)25 b(at)i(presen)n(t.)36 b(Ho)n(w)n(ev)n(er)25 b(there)h(exists)515 3353 y(an)h Fp(ad)k(ho)l(c)d FA(v)-5 b(ariational)26 b(result)h(for)h(the)f(w)n(a)n(v)n(e)f(equation)1048 3536 y Fz(w)1107 3548 y Fw(tt)1180 3536 y Fy(\000)18 b Fz(w)1322 3548 y Fw(xx)1425 3536 y FA(=)23 b Fy(\006)p Fz(w)1639 3502 y Fw(p)1696 3536 y FA(+)18 b(higher)27 b(order)f(terms)h Fz(;)97 b(p)23 b Fy(\025)g FA(2)k Fz(;)279 b FA(\(10.17\))515 3718 y(whic)n(h)23 b(ensures)f(that,)i(ha)n(ving)e (\014xed)h Fz(j)28 b Fy(\025)22 b FA(1,)i(there)f(exists)f(a)h (sequence)f(of)h(p)r(erio)r(dic)g(orbits)515 3818 y(accum)n(ulating)k (at)g(zero,)g(whose)g(frequencies)h(accum)n(ulate)f(at)h Fz(j)k FA(\(whic)n(h)c(pla)n(ys)f(here)h(the)515 3918 y(role)g(of)g(the)i Fz(j)5 b Fy(\000)p FA(th)28 b(linear)h (frequency\).)40 b(Corresp)r(onding)27 b(theorem)h(is)h(due)g(to)g (Berti)f(and)515 4017 y(Bolle)f([BB02)n(].)639 4117 y(P)n(erio)r(dic)g (solutions)f(in)i(the)g(nonlinear)f(w)n(a)n(v)n(e)f(equation)1083 4300 y Fz(w)1142 4312 y Fw(tt)1215 4300 y Fy(\000)18 b Fz(w)1357 4312 y Fw(xx)1455 4300 y FA(+)g Fz(f)9 b FA(\()p Fz(x;)14 b(w)r FA(\))25 b(=)d(0)27 b Fz(;)97 b(u)p FA(\(0)p Fz(;)14 b(t)p FA(\))23 b(=)f Fz(u)p FA(\()p Fz(\031)s(;)14 b(t)p FA(\))24 b(=)e(0)314 b(\(10.18\))515 4482 y(where)32 b(constructed)h(for)g(the)h(\014rst)e(time)i(b)n(y)f (Rabino)n(witz)g([Rab78)o(])g(using)g(global)f(v)-5 b(ari-)515 4582 y(ational)31 b(metho)r(ds)h(and)g(a)g(Ly)n(apuno)n(v{Sc)n(hmidt)e (decomp)r(osition.)50 b(Rabino)n(witz)32 b(pro)n(v)n(ed)515 4682 y(that,)g(under)f(suitable)g(assumptions)f(on)h Fz(f)9 b FA(,)32 b(equation)f(\(10.18\))e(has)i(at)g(least)g(one)f(p)r (eri-)515 4781 y(o)r(dic)k(solution)h(with)g(p)r(erio)r(d)g Fz(T)46 b FA(=)34 b(2)p Fz(\031)s(p=q)s FA(,)i(for)e(an)n(y)g(c)n (hoice)g(of)h(the)g(in)n(tegers)f Fz(p)g FA(and)h Fz(q)s FA(.)515 4881 y(Note)g(that,)i(when)e(the)h(p)r(erio)r(d)f Fz(T)46 b FA(is)35 b(commensurable)f(with)h(2)p Fz(\031)s FA(,)i(the)f(op)r(erator)d Fz(L)3254 4893 y Fw(!)3301 4881 y Fy(j)3324 4893 y Fw(R)515 4980 y FA(has)26 b(a)g(compact)g(in)n (v)n(erse,)f(i.e.)37 b(there)26 b(are)g(no)g(small)g(denominators.)35 b(The)27 b(w)n(ork)e([Rab78)o(])1905 5255 y(40)p eop PStoPSsaved restore %%Page: (40,41) 21 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 41 40 bop 515 523 a FA(w)n(as)35 b(follo)n(w)n(ed)h(b)n(y)g(a)g(series) g(of)g(pap)r(ers,)i(simplifying)f(the)g(pro)r(of)f(and)g(sharp)r(ening) g(the)515 623 y(result)26 b(\(see)h([Bre83)n(])g(and)f(references)g (therein\).)37 b(In)27 b(particular,)e(w)n(e)i(men)n(tion)f(the)i(pap)r (er)515 722 y([BCN80)o(])h(b)n(y)h(Brezis,)f(Coron)f(and)h(Niren)n(b)r (erg,)g(where)g(existence)h(of)f(p)r(erio)r(dic)g(orbits)g(is)515 822 y(pro)n(v)n(ed)c(b)n(y)i(a)f(particularly)g(simple)h(metho)r(d:)37 b(the)27 b(authors)f(write)h(a)f(v)-5 b(ariational)26 b(princi-)515 922 y(ple,)32 b(dual)e(to)h(the)g(usual)g(one,)g(and)g (lo)r(ok)f(for)h(its)g(critical)f(p)r(oin)n(ts,)i(using)e(the)h(moun)n (tain)515 1021 y(pass)j(lemma.)61 b(It)36 b(is)f(remark)-5 b(able)34 b(that)i(in)f(this)h(approac)n(h)e(the)h(Q)h(equation)e(b)r (ecomes)515 1121 y(trivial.)515 1349 y Fs(10.4)112 b(W)-9 b(eak)m(ening)37 b(the)h(nonresonance)h(condition)515 1503 y FA(The)30 b(main)g(limitation)g(of)h(the)f(results)g(presen)n (ted)f(in)i(Sections)f(10.2)f(and)h(10.3)e(rests)i(in)515 1602 y(the)f(nonresonance)f(conditions)h(H1)g(and)g(H1R.)g(Indeed,)h (suc)n(h)f(conditions)g(are)f(ful\014lled)515 1702 y(with)c(large)e (probabilit)n(y)g(\(in)j(a)e(suitable)g(parameter)f(space\))h(when)h Fz(!)2723 1714 y Fw(j)2781 1702 y Fy(\030)e Fz(j)2907 1672 y Fw(\027)2972 1702 y FA(with)i Fz(\027)29 b(>)22 b FA(1;)515 1801 y(when)29 b Fz(\027)h FA(=)25 b(1)k(the)g (nonresonance)e(conditions)h(are)g(satis\014ed)h(t)n(ypically)f(on)h (uncoun)n(table)515 1901 y(sets)40 b(of)g(zero)g(measure,)j(but)e(when) f Fz(\027)50 b(<)44 b FA(1)d(they)f(are)g(satis\014ed)g(only)g (exceptionally)515 2001 y(\(as)33 b(in)g(the)h(plate)f(equation\).)53 b(As)34 b(a)f(consequence)f(the)h(results)g(of)g(Sections)g(10.2)f(and) 515 2100 y(10.3)23 b(are)g(not)h(applicable)f(to)h(general)f(equations) g(in)i(more)e(than)h(one)g(space)f(dimensions.)515 2200 y(F)-7 b(urthermore,)22 b(the)h(metho)r(d)g(of)f(Ly)n(apuno)n(v{Sc)n (hmidt)f(decomp)r(osition)h(can)g(b)r(e)h(extended)515 2300 y(to)18 b(the)h(case)f(of)h(rev)n(ersible)e(systems)h(of)h (\014rst)g(order)e(in)i(time,)i(but)e(the)h(approac)n(h)c(of)j(Section) 515 2399 y(10.2)26 b(is)h(no)h(more)f(applicable.)639 2499 y(In)e(order)e(to)h(a)n(v)n(oid)e(suc)n(h)i(limitations)h(one)f(w) n(ould)f(lik)n(e)h(to)g(b)r(e)h(able)f(to)g(w)n(ork)f(with)i(the)515 2598 y(w)n(eak)n(er)19 b(nonresonance)h(condition)h(\\there)g(exists)g (a)g Fz(\034)33 b(>)22 b FA(0)f(suc)n(h)g(that)h Fy(j)p Fz(l)r(!)9 b Fy(\000)d Fz(!)2990 2610 y Fw(j)3024 2598 y Fy(j)23 b(\025)f Fz(\015)5 b(=l)3274 2568 y Fw(\034)3314 2598 y FA(".)515 2698 y(This)24 b(w)n(as)g(done)g(b)n(y)h(Craig)e(and)i (W)-7 b(a)n(yne)24 b([CW93)o(])h(who)f(used)h(the)g(Nash{Moser)e (theorem)515 2798 y(to)g(solv)n(e)f(the)i Fz(P)35 b FA(equation.)g(The) 24 b(application)f(of)g(the)h(Nash)f(Moser)f(theorem)h(requires)f(to) 515 2897 y(construct)k(and)h(estimate)g(the)h(in)n(v)n(erse)d(of)i(the) h(linear)e(op)r(erator)f(describing)i(the)g(lineari-)515 2997 y(sation)e(of)g(the)h(P)f(equation)g(at)g(an)g(appro)n(ximate)f (solution.)36 b(This)25 b(is)g(the)h(main)g(di\016cult)n(y)515 3097 y(of)h(Craig{W)-7 b(a)n(yne's)25 b(approac)n(h.)36 b(T)-7 b(o)27 b(o)n(v)n(ercome)e(it)k(they)e(use)h(the)g(tec)n(hniques) f(b)n(y)h(F)-7 b(r\177)-42 b(olic)n(h)515 3196 y(and)36 b(Sp)r(encer)g([FS83)o(],)j(p)r(erforming)c(a)h(careful)g(analysis)f (of)h(small)g(denominators)e(\(cf.)515 3296 y(Section)e(5.3\).)52 b(The)33 b(metho)r(d)g(b)n(y)g(Craig)e(and)h(W)-7 b(a)n(yne)33 b(w)n(as)e(extended)i(b)n(y)g(Bourgain)e(in)515 3395 y(order)25 b(to)i(construct)f(p)r(erio)r(dic)g(\(and)h(also)e(quasi)h (p)r(erio)r(dic\))h(solutions)f(in)h(higher)f(dimen-)515 3495 y(sional)34 b(equations.)58 b(The)35 b(resulting)f(metho)r(d)i (seems)e(v)n(ery)g(general,)i(but)f(at)g(presen)n(t)f(a)515 3595 y(theorem)h(\\ready)g(for)g(application")g(is)h(not)g(a)n(v)-5 b(ailable.)61 b(W)-7 b(e)37 b(presen)n(t)e(here)h(the)g(result)515 3694 y(obtained)27 b(b)n(y)g(Bourgain)f(b)n(y)i(applying)f(this)g (metho)r(d)i(to)e(the)h(nonlinear)f(w)n(a)n(v)n(e)f(equation)1494 3854 y Fz(w)1553 3866 y Fw(tt)1626 3854 y Fy(\000)18 b FA(\001)p Fz(w)k FA(+)c Fz(aw)j FA(+)d Fz(w)2210 3820 y Fv(3)2271 3854 y FA(=)k(0)725 b(\(10.19\))515 4014 y(on)27 b Fx(T)686 3984 y Fw(d)724 4014 y FA(.)37 b(Fix)28 b(a)f(m)n(ultiindex)h Fz(n)23 b Fy(2)g Fx(Z)1630 3984 y Fw(d)1690 4014 y FA(di\013eren)n(t)28 b(from)f(zero,)g(and)g(let)882 4206 y Fz(\030)918 4218 y Fw(n)964 4206 y FA(\()p Fz(!)1048 4218 y Fw(n)1093 4206 y Fz(t;)14 b(x)p FA(\))23 b(:=)g(cos)o(\()p Fz(n)c Fy(\001)f Fz(x)h FA(+)f Fz(!)1827 4218 y Fw(n)1872 4206 y Fz(t)p FA(\))28 b Fz(;)97 b(!)2134 4218 y Fw(n)2202 4206 y FA(:=)2313 4106 y Fq(q)p 2396 4106 594 4 v 100 x Fz(n)2446 4177 y Fv(2)2446 4228 y(1)2501 4206 y FA(+)18 b Fz(:::)h FA(+)f Fz(n)2805 4177 y Fv(2)2805 4231 y Fw(d)2862 4206 y FA(+)g Fz(a;)515 4379 y FA(b)r(e)28 b(the)g(corresp)r(onding)d (symmetric)j(rev)n(ersible)e(solution.)515 4527 y FB(Theorem)34 b(10.8.)42 b Fp([se)l(e)34 b([Bou95b)r(]].)49 b(If)34 b Fz(a)f Fp(b)l(elongs)g(to)g(a)h(c)l(ertain)f(subset)f(of)i Fx(R)3067 4496 y Fv(+)3161 4527 y Fp(of)g(ful)t(l)515 4626 y(me)l(asur)l(e,)d(then)g(ther)l(e)g(exists)g(a)g(Cantor)h(set)f Fy(E)38 b Fp(of)32 b(p)l(ositive)g(me)l(asur)l(e,)g(ac)l(cumulating)f (at)515 4726 y(zer)l(o,)c(and)f(a)g(family)i(of)f(p)l(erio)l(dic)g (solutions)f Fy(f)p Fz(w)2031 4738 y Fw(\017)2063 4726 y FA(\()p Fz(t;)14 b(x)p FA(\))p Fy(g)2283 4738 y Fw(\017)p Fu(2E)2427 4726 y Fp(of)26 b(\(10.19\))i(with)e(fr)l(e)l(quencies)515 4825 y Fz(!)570 4795 y Fw(\017)601 4825 y Fp(,)k(satisfying)1033 4985 y Fy(j)p Fz(\017\030)1126 4997 y Fw(n)1171 4985 y FA(\()p Fz(!)1258 4951 y Fw(\017)1290 4985 y Fz(t;)14 b(x)p FA(\))19 b Fy(\000)f Fz(w)1597 4997 y Fw(\017)1630 4985 y FA(\()p Fz(t;)c(x)p FA(\))p Fy(j)24 b(\024)e Fz(C)6 b(\017)2041 4951 y Fv(3)2108 4985 y Fz(;)99 b Fy(j)p Fz(!)2305 4997 y Fw(n)2368 4985 y Fy(\000)18 b Fz(!)2506 4951 y Fw(\017)2538 4985 y Fy(j)23 b(\024)g Fz(C)6 b(\017)2771 4951 y Fv(2)2837 4985 y Fz(:)1905 5255 y FA(41)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 42 41 bop 639 523 a FA(In)25 b(the)g(case)f Fz(d)f FA(=)g(1,)h(the)h (result)g(w)n(as)e(pro)n(v)n(ed)g(in)i([CW93)o(];)h(subsequen)n(tly)-7 b(,)25 b(still)g(in)g(the)515 623 y(case)30 b Fz(d)g FA(=)g(1,)i(Kuksin)f(in)n(tro)r(duced)g(a)g(simpler)h(tec)n(hnique)f (to)h(\014nd)g(the)f(\\large)f(measure)515 722 y(result")d(of)g (theorem)g(10.8)f(\(see)i(in)g([Bou99a)n(])g(pp.)g(90{94\).)639 822 y(The)k(Craig{W)-7 b(a)n(yne{Bourgain)28 b(metho)r(d)k(also)f(allo) n(ws)g(to)h(deal)g(with)g(\014rst)g(order)f(in)515 922 y(time)26 b(equations.)36 b(F)-7 b(or)25 b(example,)h(it)g(w)n(as)f (applied)h(to)g(the)g(Sc)n(hr\177)-42 b(odinger)24 b(equation)h(in)h (one)515 1021 y([CW94)o(])i(or)f(t)n(w)n(o)f(space)h(dimensions)g ([Bou98)o(].)515 1247 y Fs(10.5)112 b(The)38 b(w)m(ater)f(w)m(a)m(v)m (e)h(problem)515 1400 y FA(A)24 b(particular)f(problem)g(that)h(has)g (attracted)f(the)h(atten)n(tion)g(of)g(man)n(y)f(researc)n(hers)e (since)515 1499 y(the)27 b(v)n(ery)e(b)r(eginning)h(of)h(the)g(theory)e (of)i(PDEs)f(is)g(that)h(of)f(existence)h(of)f(standing)g(w)n(ater)515 1599 y(w)n(a)n(v)n(es.)34 b(The)27 b(\014rst)f(rigorous)e(pro)r(of)i (of)g(their)h(existence)f(w)n(as)f(obtained)i(only)f(recen)n(tly)f(b)n (y)515 1699 y(Plotnik)n(o)n(v)g(and)j(T)-7 b(oland)27 b([PT01)o(];)g(w)n(e)h(presen)n(t)f(here)g(their)g(result.)639 1798 y(Consider)j(a)h(p)r(erfect)h(\015uid)f(lying)g(ab)r(o)n(v)n(e)f (a)h(horizon)n(tal)e(b)r(ottom,)k(and)e(con\014ned)g(b)r(e-)515 1898 y(t)n(w)n(een)19 b(t)n(w)n(o)g(parallel)f(v)n(ertical)h(w)n(alls.) 33 b(The)20 b(\015uid)g(is)g(sub)5 b(ject)20 b(to)f(gra)n(vit)n(y)-7 b(,)20 b(and)f(atmospheric)515 1998 y(pressure)f(acts)i(at)g(the)g (free)g(surface.)34 b(This)20 b(is)g(a)f(dynamical)h(system)g(go)n(v)n (erned)d(b)n(y)j(the)h(Eu-)515 2097 y(ler)h(equations)g(supplemen)n (ted)h(b)n(y)g(appropriate)e(b)r(oundary)h(conditions.)35 b(It)23 b(w)n(as)f(p)r(oin)n(ted)515 2197 y(out)f(b)n(y)h(Zakharo)n(v)d (that)j(this)g(system)f(is)h(Hamiltonian)f(\(see)h([Zak68)n(]\).)35 b(The)22 b(corresp)r(ond-)515 2297 y(ing)g(Hamiltonian)h(function)g(is) g(the)g(energy)f(of)g(the)h(\015uid,)i(and)d(conjugated)h(v)-5 b(ariables)21 b(are)515 2396 y(giv)n(en)27 b(b)n(y)g(the)h(w)n(a)n(v)n (e)e(pro\014le)h(and)g(the)h(v)n(elo)r(cit)n(y)f(p)r(oten)n(tial)g(at)h (the)g(free)f(surface.)639 2496 y(In)21 b(the)g(linear)e(appro)n (ximation)g(the)i(general)e(solution)h(is)g(giv)n(en)g(b)n(y)g(the)h (sup)r(erp)r(osition)515 2595 y(of)26 b(the)h(normal)f(mo)r(des.)36 b(The)27 b(problem)f(is)h(to)f(con)n(tin)n(ue)g(the)h(normal)f(mo)r (des)h(to)f(families)515 2695 y(of)31 b(p)r(erio)r(dic)f(solutions)h (of)f(the)i(nonlinear)e(system)g(\(the)i(standing)e(w)n(a)n(v)n(es\).) 46 b(Fix)31 b(one)f(of)515 2795 y(the)g(normal)f(mo)r(des,)h(and)g (denote)g(b)n(y)f Fz(\021)s FA(\()p Fz(t;)14 b(x)1967 2807 y Fv(1)2005 2795 y FA(\))31 b(the)f(corresp)r(onding)e(pro\014le)h (of)h(the)g(free)515 2894 y(surface)36 b(\()p Fz(x)883 2906 y Fv(1)958 2894 y FA(b)r(eing)h(the)g(horizon)n(tal)f(v)-5 b(ariable\).)64 b(Then)37 b(it)h(is)e(p)r(ossible)h(to)g(c)n(ho)r(ose)f (the)515 2994 y(depth)c Fz(h)p FA(,)g(the)g(width)g Fz(l)h FA(of)e(the)h(region)e(o)r(ccupied)i(b)n(y)f(the)h(\015uid)f(and)h(the) g(gra)n(vitational)515 3094 y(constan)n(t)i Fz(g)j FA(in)e(suc)n(h)g(a) f(w)n(a)n(y)f(that)i(the)h(p)r(erio)r(d)e(of)h(the)g(solution)f(is)h (normalised)f(to)g(2)p Fz(\031)515 3193 y FA(and)c(the)g(linear)f (frequencies)h(ful\014ll)h(a)e(suitable)h(nonresonance)e(condition.)44 b(Denote)30 b(b)n(y)515 3293 y(\()p Fz(g)587 3305 y Fv(0)624 3293 y Fz(;)14 b(l)686 3305 y Fv(0)723 3293 y Fz(;)g(h)808 3305 y Fv(0)845 3293 y FA(\))28 b(a)f(c)n(hoice)g(of)g(the)h (parameters)e(realizing)g(suc)n(h)i(conditions,)f(then)h(one)f(has)515 3427 y FB(Theorem)38 b(10.9.)44 b Fp([se)l(e)37 b([PT01)r(]])g(Ther)l (e)g(exists)e(an)i(in\014nite)e(set)h Fy(E)42 b(\032)34 b Fx(R)42 b Fp(having)c(zer)l(o)515 3526 y(as)33 b(an)f(ac)l (cumulation)h(p)l(oint)g(and,)h(for)g(any)f Fz(\017)28 b Fy(2)g(E)7 b Fp(,)34 b(ther)l(e)f(exist)f Fz(g)2657 3538 y Fw(\017)2689 3526 y Fp(,)h Fz(l)2772 3538 y Fw(\017)2837 3526 y Fp(and)g(a)g(standing)515 3626 y(wave)k(solution)f(of)h(the)f (water)g(wave)h(pr)l(oblem)g(with)g(gr)l(avity)g Fz(g)2560 3638 y Fw(\017)2627 3626 y Fp(in)f(a)g(b)l(ox)g(of)h(width)g Fz(l)3322 3638 y Fw(\017)3354 3626 y Fp(.)515 3725 y(Mor)l(e)l(over,)31 b(denoting)g(by)f Fz(\021)1392 3737 y Fw(\017)1454 3725 y Fp(the)g(c)l(orr)l(esp)l(onding)h(pr)l(o\014le)f(of)h(the)e(fr)l(e)l (e)i(surfac)l(e,)f(one)g(has)901 3868 y Fy(j)p Fz(\021)965 3880 y Fw(\017)997 3868 y FA(\()p Fz(t;)14 b(x)1143 3880 y Fv(1)1181 3868 y FA(\))k Fy(\000)g Fz(\017)1348 3833 y Fv(2)1385 3868 y Fz(\021)s FA(\()p Fz(t;)c(x)1575 3880 y Fv(1)1613 3868 y FA(\))p Fy(j)24 b Fz(<)e(C)6 b(\017)1878 3833 y Fv(3)1945 3868 y Fz(;)99 b Fy(j)p Fz(g)2130 3880 y Fw(\017)2180 3868 y Fy(\000)18 b Fz(g)2303 3880 y Fv(0)2340 3868 y Fy(j)g FA(+)g Fy(j)p Fz(l)2512 3880 y Fw(\017)2563 3868 y Fy(\000)g Fz(l)2671 3880 y Fv(0)2708 3868 y Fy(j)23 b(\024)f Fz(C)6 b(\017)30 b(:)639 4010 y FA(The)g(main)g(di\016culties) g(in)g(pro)n(ving)e(this)i(result)g(are)e(as)h(follo)n(ws:)41 b(\014rstly)-7 b(,)30 b(the)g(linear)515 4110 y(frequencies)i(b)r(eha)n (v)n(e)h(as)f Fz(!)1387 4122 y Fw(n)1464 4110 y Fy(\030)g Fz(n)1611 4079 y Fv(1)p Fw(=)p Fv(2)1716 4110 y FA(,)i(so)f(the)g (nonresonance)f(conditions)g(that)i(can)f(b)r(e)515 4209 y(satis\014ed)e(are)g(quite)g(w)n(eak.)48 b(Secondly)-7 b(,)33 b(the)f(mathematical)f(form)n(ulation)f(of)i(the)g(prob-)515 4309 y(lem)41 b(in)n(v)n(olv)n(es)e(an)h(un)n(b)r(ounded)h(nonlinear)f (and)g(non-lo)r(cal)g(op)r(erator.)74 b(T)-7 b(o)40 b(o)n(v)n(ercome) 515 4408 y(these)33 b(di\016culties,)j(Plotnik)n(o)n(v)31 b(and)j(T)-7 b(oland)32 b(use)i(the)g(Lagrangian)c(description)j(of)h (the)515 4508 y(\015uid)23 b(motion)g(and)g(apply)g(the)g(Ly)n(apuno)n (v{Sc)n(hmidt)e(approac)n(h)g(to)i(handle)g(the)h(resulting)515 4608 y(nonlinear)i(problem.)36 b(The)27 b(P)g(equation)f(no)n(w)g(is)h (solv)n(ed)f(b)n(y)h(means)g(of)g(the)g(Nash{Moser)515 4707 y(theorem.)35 b(The)25 b(required)f(in)n(v)n(ertibilit)n(y)g(of)g (the)h(linearised)f(op)r(erator)f(is)i(obtained)f(in)h(t)n(w)n(o)515 4807 y(steps:)46 b(\014rst)32 b(it)g(is)g(reduced)g(to)g(a)g(suitable)g (canonical)f(form,)i(and)f(next)h(this)f(canonical)515 4907 y(form)j(\(whic)n(h)g(is)g(essen)n(tially)f(a)h(p)r(erturbation)f (of)h(an)g(op)r(erator)f(in)n(v)n(olving)f(deriv)-5 b(ativ)n(es)515 5006 y(and)27 b(Hilb)r(ert)h(transform\))f(is)h(studied)g(in)f(detail.) 1905 5255 y(42)p eop PStoPSsaved restore %%Page: (42,43) 22 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 43 42 bop 515 523 a FC(References)515 705 y FA([Arn63])159 b(V.)27 b(I.)f(Arnold,)h Fp(Pr)l(o)l(of)j(of)f(a)g(the)l(or)l(em)g(of)h (A.N.Kolmo)l(gor)l(ov)g(on)f(the)g(c)l(onser-)945 805 y(vation)g(of)g(quasip)l(erio)l(dic)h(motions)f(under)f(a)g(smal)t(l)h (change)g(of)g(the)g(Hamil-)945 904 y(tonian)h(function)p FA(,)e(Russ.)f(Math.)h(Surv.)f FB(18)g FA(\(1963\),)g(no.)g(5,)g(9{36.) 515 1070 y([Arn89])159 b(V.)26 b(I.)f(Arnold,)g Fp(Mathematic)l(al)30 b(metho)l(ds)e(in)g(classic)l(al)h(me)l(chanics)p FA(,)e(3rd)e(ed.,)945 1170 y(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1989.)515 1336 y([Bam99a])88 b(D.)29 b(Bam)n(busi,)f Fp(Nekhor)l(oshev)33 b(the)l(or)l(em)d(for)i(smal)t(l)f(amplitude)h(solutions)f(in)945 1435 y(nonline)l(ar)f(Schr\177)-42 b(odinger)31 b(e)l(quation)p FA(,)d(Math.)g(Z.)g FB(130)f FA(\(1999\),)f(345{387.)515 1602 y([Bam99b])84 b(D.)26 b(Bam)n(busi,)g Fp(On)h(long)i(time)f (stability)h(in)f(Hamiltonian)h(p)l(erturb)l(ations)f(of)945 1701 y(non-r)l(esonant)h(line)l(ar)h(PDEs)p FA(,)e(Nonlinearit)n(y)g FB(12)f FA(\(1999\),)f(823{850.)515 1867 y([Bam00])130 b(D.)26 b(Bam)n(busi,)f Fp(Lyapunov)k(c)l(enter)e(the)l(or)l(em)h(for)g (some)h(nonline)l(ar)f(PDEs:)38 b(a)945 1967 y(simple)31 b(pr)l(o)l(of)p FA(,)e(Ann.)f(S.N.S.)g(Pisa)f(Cl.)h(Sci)f FB(29)g FA(\(2000\),)g(823{837.)515 2133 y([BB02])182 b(M.)19 b(Berti)f(and)h(P)-7 b(.)18 b(Bolle,)i Fp(Perio)l(dic)k (solutions)e(of)g(nonline)l(ar)g(wave)h(e)l(quations)945 2232 y(with)30 b(gener)l(al)h(nonline)l(arities)p FA(,)d(Preprin)n(t)f (\(2002\).)515 2399 y([BBE)713 2368 y Fv(+)767 2399 y FA(94])71 b(E.)26 b(D.)g(Belok)n(olos,)e(A.)j(I.)f(Bob)r(enk)n(o,)f(V.) i(Z.)f(Enolskii,)f(A.)i(R.)f(Its,)h(and)f(V.)g(B.)945 2498 y(Matv)n(eev,)40 b Fp(A)n(lgebr)l(o-ge)l(ometric)g(appr)l(o)l(ac)i (to)d(nonline)l(ar)h(inte)l(gr)l(able)g(e)l(qua-)945 2598 y(tions)p FA(,)28 b(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1994.) 515 2764 y([BCN80])119 b(H.)26 b(Brezis,)g(J.)g(Coron,)f(and)h(L.)h (Niren)n(b)r(erg,)e Fp(F)-6 b(r)l(e)l(e)28 b(vibr)l(ations)i(for)f(a)g (nonlin-)945 2863 y(e)l(ar)g(wave)i(e)l(quation)e(and)h(a)g(the)l(or)l (em)f(by)h(P.)g(R)l(abinowitz)p FA(,)e(Comm)n(un.)f(Pure)945 2963 y(Appl.)h(Math.)g FB(33)f FA(\(1980\),)f(667{689.)515 3129 y([BF)n(G98])124 b(G.)34 b(Benettin,)h(F.)f(F)-7 b(asso,)33 b(and)h(M.)f(Guzzo,)i Fp(Nekhor)l(oshev)i(stability)f(of)g (el-)945 3229 y(liptic)c(e)l(quilibr)l(a)h(of)f(Hamiltonian)g(systems)p FA(,)d(Comm.)h(Math.)f(Ph)n(ysics)f FB(197)945 3328 y FA(\(1998\),)e(347{360.)515 3494 y([BK91])176 b(A.)24 b(I.)g(Bob)r(enk)n(o)e(and)i(S.)f(B.)h(Kuksin,)g Fp(Finite-gap)j(p)l (erio)l(dic)i(solutions)d(of)h(the)945 3594 y(KdV)32 b(e)l(quation)g(ar)l(e)g(nonde)l(gener)l(ate)p FA(,)f(Ph)n(ys.)e(Lett.) i(A)f FB(161)f FA(\(1991\),)h(no.)f(3,)945 3694 y(274{276.)515 3860 y([BK93])176 b(R.)24 b(F.)g(Bikbaev)g(and)f(S.)i(B.)f(Kuksin,)g Fp(A)i(p)l(erio)l(dic)i(b)l(oundary-value)g(pr)l(oblem)945 3959 y(for)38 b(the)f(Sine-Gor)l(don)h(e)l(quation,)h(smal)t(l)f (Hamiltonian)g(p)l(erturb)l(ations)f(of)945 4059 y(it,)i(and)f (KAM-defomations)h(of)f(\014nite-gap)f(tori)p FA(,)h(St.-P)n(etersburg) c(Math.)945 4159 y(J.)27 b FB(4)h FA(\(1993\),)e(439{468.)515 4325 y([BK95a])134 b(A.)25 b(I.)f(Bob)r(enk)n(o)g(and)g(S.)h(B.)f (Kuksin,)h Fp(The)j(nonline)l(ar)f(Klein-Gor)l(don)h(e)l(qua-)945 4424 y(tion)e(on)h(an)f(interval)h(as)g(a)g(p)l(erturb)l(e)l(d)f (Sine-Gor)l(don)h(e)l(quation)p FA(,)e(Commen)n(t.)945 4524 y(Math.)j(Helv.)f FB(70)h FA(\(1995\),)e(63{112.)515 4690 y([BK95b])130 b(A.)30 b(I.)g(Bob)r(enk)n(o)f(and)g(S.)h(B.)g (Kuksin,)g Fp(Smal)t(l-)i(amplitude)h(solutions)f(of)h(the)945 4790 y(Sine-Gor)l(don)i(e)l(quation)g(on)f(an)h(interval)g(under)f (Dirichlet)i(or)f(Neumann)945 4889 y(b)l(oundary)30 b(c)l(onditions)p FA(,)f(J.)f(Nonlinear)e(Sci.)i FB(5)f FA(\(1995\),)g(207{232.)1905 5255 y(43)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 44 43 bop 515 523 a FA([Bou93])153 b(J.)23 b(Bourgain,)f Fp(F)-6 b(ourier)25 b(tr)l(ansform)h(r)l(estriction)g(phenomenona)g (for)h(c)l(ertain)945 623 y(lattic)l(e)c(subsets)g(and)h(applic)l (ations)h(to)e(nonline)l(ar)h(evolution)g(e)l(quations)p FA(,)e(Ge-)945 722 y(ometric)27 b(and)g(F)-7 b(unctional)28 b(Analysis)f FB(3)g FA(\(1993\),)g(107{156)d(and)j(209{262.)515 883 y([Bou94])153 b(J.)33 b(Bourgain,)f Fp(Construction)j(of)h(quasi-p) l(erio)l(dic)h(solutions)e(for)h(Hamilto-)945 983 y(nian)28 b(p)l(erturb)l(ations)g(of)i(line)l(ar)e(e)l(quations)h(and)g(applic)l (ations)h(to)e(nonline)l(ar)945 1082 y(PDE)p FA(,)g(In)n(ternat.)f (Math.)h(Res.)f(Notices)h(\(1994\),)e(475{497.)515 1243 y([Bou95a])111 b(J.)39 b(Bourgain,)i Fp(Asp)l(e)l(cts)g(of)g(long)h (time)f(b)l(ehaviour)h(of)g(solutions)f(of)h(non-)945 1343 y(line)l(ar)35 b(Hamiltonian)g(evolution)g(e)l(quations)p FA(,)f(Geometric)f(and)f(F)-7 b(unctional)945 1443 y(Analysis)27 b FB(5)g FA(\(1995\),)g(105{140.)515 1604 y([Bou95b])107 b(J.)35 b(Bourgain,)h Fp(Construction)h(of)h(p)l(erio)l(dic)h (solutions)e(of)h(nonline)l(ar)f(wave)945 1703 y(e)l(quations)32 b(in)g(higher)h(dimension)p FA(,)f(Geometric)d(and)h(F)-7 b(unctional)30 b(Analysis)945 1803 y FB(5)d FA(\(1995\),)g(629{639.)515 1964 y([Bou98])153 b(J.)29 b(Bourgain,)g Fp(Quasi-p)l(erio)l(dic)k (solutions)f(of)g(Hamiltonian)h(p)l(erturb)l(ations)945 2063 y(of)d(2D)g(line)l(ar)h(Sh\177)-42 b(odinger)31 b(e)l(quation)p FA(,)d(Ann.)g(Math.)g FB(148)f FA(\(1998\),)f(363{439.) 515 2224 y([Bou99a])111 b(J.)20 b(Bourgain,)f Fp(Nonline)l(ar)24 b(Schr\177)-42 b(odinger)24 b(e)l(quations)p FA(,)e(Hyp)r(erb)r(olic)e (equations)945 2324 y(and)27 b(frequency)g(in)n(teractions,)g(American) g(Mathematical)g(So)r(ciet)n(y)-7 b(,)28 b(1999.)515 2485 y([Bou99b])107 b(J.)64 b(Bourgain,)72 b Fp(Perio)l(dic)66 b(solutions)e(of)g(nonline)l(ar)g(wave)h(e)l(quations)p FA(,)945 2585 y(Harmonic)60 b(analysis)g(and)i(partial)e(di\013eren)n (tial)h(equations,)69 b(Chicago)945 2684 y(Univ.Press,)26 b(1999,)g(pp.)i(69{97.)515 2845 y([Bou00])153 b(J.)24 b(Bourgain,)f Fp(On)j(di\013usion)i(in)f(high-dimensional)i (Hamiltonian)f(systems)945 2945 y(and)i(PDE)p FA(,)e(J.)g(Anal.)f (Math.)h FB(80)f FA(\(2000\),)f(1{35.)515 3106 y([BP01])184 b(D.)28 b(Bam)n(busi)f(and)h(S.)g(P)n(aleari,)d Fp(F)-6 b(amilies)32 b(of)e(p)l(erio)l(dic)i(orbits)f(for)f(r)l(esonant)945 3205 y(PDE's)p FA(,)e(J.)g(Nonlinear)e(Science)i FB(11)f FA(\(2001\),)g(69{87.)515 3366 y([BP02])184 b(D.)41 b(Bam)n(busi)f(and) g(S.)h(P)n(aleari,)h Fp(F)-6 b(amilies)43 b(of)f(p)l(erio)l(dic)i (orbits)e(for)h(some)945 3466 y(PDE's)30 b(in)f(higher)h(dimensions)p FA(,)f(Comm)n(un.)e(on)f(Pure)g(and)h(Applied)g(Anal-)945 3565 y(ysis)g FB(1)g FA(\(2002\),)g(269{279.)515 3726 y([Bre83])171 b(H.)26 b(Brezis,)f Fp(Perio)l(dic)30 b(solutions)e(of)h (nonline)l(ar)g(vibr)l(ating)f(string)g(and)h(dual-)945 3826 y(ity)h(principle)p FA(,)g(Bull.)d(A.M.S.)i FB(8)e FA(\(1983\),)g(409{426.)515 3987 y([CMM02])88 b(D.)27 b(Cai,)f(D.)g(W.)h(McLaughlin,)f(and)g(K.)g(T.)g(R.)h(McLaughlin,)f Fp(The)j(Nonlin-)945 4087 y(e)l(ar)35 b(Schr\177)-42 b(odinger)37 b(e)l(quation)e(as)g(b)l(oth)g(a)h(PDE)f(and)g(a)g (dynamic)l(al)i(system)p FA(,)945 4186 y(Handb)r(o)r(ok)28 b(of)g(Dynamical)h(Systems)f(\(B.)h(Fiedler,)f(ed.\),)h(v)n(ol.)f(2,)g (Elsevier,)945 4286 y(Amsterdam,)f(2002.)515 4447 y([Cra00])165 b(W.)65 b(Craig,)73 b Fp(Pr)l(obl)n(\022)-40 b(emes)65 b(de)f(p)l(etitts)g(diviseurs)h(dans)f(les)e(\023)-40 b(equations)945 4546 y(aux)47 b(d)n(\023)-40 b(eriv)n(\023)g(ees)50 b(p)l(artiel)t(les)p FA(,)k(P)n(anoramas)44 b(et)j(Syn)n(th)n(\023)-39 b(eses,)51 b(no.)c(9,)52 b(So)r(ci)n(\023)-39 b(et)n(\023)g(e)945 4646 y(Math)n(\023)g(ematique)26 b(de)i(F)-7 b(rance,)27 b(2000.)515 4807 y([CW93])155 b(W.)25 b(Craig)d(and)i(C.)h(E.)f(W)-7 b(a)n(yne,)24 b Fp(Newton)-8 b('s)27 b(metho)l(d)g(and)g(p)l(erio)l (dic)i(solutions)945 4907 y(of)34 b(nonline)l(ar)g(wave)g(e)l(quations) p FA(,)f(Comm.)e(Pure)g(Appl.)h(Math.)f FB(46)g FA(\(1993\),)945 5006 y(1409{1498.)1905 5255 y(44)p eop PStoPSsaved restore %%Page: (44,45) 23 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 45 44 bop 515 523 a FA([CW94])155 b(W.)59 b(Craig)f(and)h(C.)g(E.)f(W) -7 b(a)n(yne,)67 b Fp(Perio)l(dic)60 b(solutions)f(of)g(nonline)l(ar) 945 623 y(Schr\177)-42 b(odinger)45 b(e)l(quations)f(and)g(the)g (Nash-Moser)h(metho)l(d)p FA(,)i(Hamiltonian)945 722 y(mec)n(hanics.)g(In)n(tegrabilit)n(y)f(and)i(c)n(haotic)e(b)r(eha)n (vior,)52 b(NA)-7 b(TO)48 b(ASI,)g(v)n(ol.)945 822 y(B331,)26 b(Plen)n(um)h(Press,)f(1994,)g(pp.)i(103{122.)515 983 y([CY00])178 b(L.)36 b(Chierc)n(hia)f(and)h(J.)g(Y)-7 b(ou,)38 b Fp(KAM)g(tori)g(for)g(1D)g(nonline)l(ar)g(wave)g(e)l(qua-) 945 1082 y(tions)28 b(with)g(p)l(erio)l(dic)i(b)l(oundary)e(c)l (onditions)p FA(,)g(Comm.)d(Math.)h(Ph)n(ysics)e FB(211)945 1182 y FA(\(2000\),)i(497{525.)515 1343 y([DMN76])99 b(B.)27 b(A.)g(Dubro)n(vin,)g(V.)g(F.)g(Matv)n(eev,)g(and)f(S.P)-7 b(.)27 b(No)n(vik)n(o)n(v,)e Fp(Nonline)l(ar)30 b(e)l(qua-)945 1443 y(tions)39 b(of)g(Kortewe)l(g-de-Vries)g(typ)l(e,)j(\014nite)c (zone)h(line)l(ar)g(op)l(er)l(ators,)j(and)945 1542 y(Ab)l(elian)30 b(varieties)p FA(,)f(Russ.)f(Math.)g(Surv.)f FB(31)g FA(\(1976\),)f(no.)i(1,)f(55{135.)515 1703 y([Dub81])145 b(B.)42 b(A.)h(Dubro)n(vin,)i Fp(Theta-functions)f(and)g(nonline)l(ar)g (e)l(quations)p FA(,)i(Russ.)945 1803 y(Math.)28 b(Surv.)f FB(36)g FA(\(1981\),)f(no.)i(2,)f(11{80.)515 1964 y([EKMY02])40 b(L.)26 b(H.)g(Eliasson,)f(S.)h(B.)g(Kuksin,)g(S.)h(Marmi,)f(and)f (J.-C.)h(Y)-7 b(o)r(ccoz,)26 b Fp(Dynami-)945 2063 y(c)l(al)31 b(systems)g(and)g(smal)t(l)h(divisors)p FA(,)g(Lecture)c(Notes)h(in)g (Mathematics,)g(v)n(ol.)945 2163 y(1784,)24 b(c)n(h.)h(KAM{p)r (ersistence)f(of)i(\014nite-gap)f(solutions,)g(Springer,)f(Berlin,)945 2263 y(2002.)515 2424 y([FPU65])127 b(E.)36 b(F)-7 b(ermi,)39 b(J.R.)d(P)n(asta,)h(and)f(S.M.)h(Ulam,)i Fp(Studies)e(of)i(nonline)l (ar)f(pr)l(ob-)945 2523 y(lems)p FA(,)29 b(Collected)f(w)n(orks)e(of)j (E.)e(F)-7 b(ermi,)29 b(v)n(ol.2,)e(Chicago)g(Univ)n(ersit)n(y)h (Press,)945 2623 y(Chicago,)e(1965.)515 2784 y([F)-7 b(ri85])197 b(L.)32 b(F)-7 b(riedlender,)32 b Fp(A)n(n)h(invariant)i (me)l(asur)l(e)e(for)i(the)e(e)l(quation)h Fz(u)3005 2796 y Fw(tt)3080 2784 y Fy(\000)21 b Fz(u)3214 2796 y Fw(xx)3314 2784 y FA(+)945 2883 y Fz(u)993 2853 y Fv(3)1053 2883 y FA(=)h(0)p Fp(.)p FA(,)28 b(Comm.)g(Math.)f(Ph)n(ys.)g FB(98)g FA(\(1985\),)f(1{16.)515 3044 y([FS83])200 b(J.)29 b(F)-7 b(r\177)-42 b(ohlic)n(h)28 b(and)h(T.)g(Sp)r(encer,)h Fp(A)n(bsenc)l(e)g(of)i(di\013usion)g(in)g(Anderson)f(tight)945 3144 y(binding)42 b(mo)l(del)h(for)f(lar)l(ge)h(disor)l(der)g(or)f(low) g(ener)l(gy)p FA(,)j(Comm)n(un.)40 b(Math.)945 3244 y(Ph)n(ys.)26 b FB(88)i FA(\(1983\),)e(151{184.)515 3405 y([Gia72])170 b(G.)59 b(E.)g(Giacaglia,)65 b Fp(Perturb)l(ation)59 b(metho)l(ds)g(in)f(non-line)l(ar)h(systems)p FA(,)945 3504 y(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1972.)515 3665 y([GK03])170 b(B.)19 b(Gr)n(\023)-39 b(eb)r(ert)17 b(and)i(T.)g(Kapp)r(eler,)h Fp(Perturb)l(ation)h(of)i(the)f(defo)l (cusing)h(nls)e(e)l(qua-)945 3765 y(tion)k(with)h(p)l(erio)l(dic)h(b)l (oundary)e(c)l(onditions)p FA(,)g(Milan)d(J.)h(of)f(Math.,)i(to)e(app)r (ear)945 3864 y(\(2003\).)515 4025 y([Gro85])160 b(M.)24 b(Gromo)n(v,)e Fp(Pseudoholomorphic)31 b(curves)26 b(in)g(symple)l (ctic)h(manifolds)p FA(,)f(In-)945 4125 y(v)n(en)n(t.)h(Math.)h FB(82)f FA(\(1985\),)f(307{347.)515 4286 y([HZ94])187 b(H.)36 b(Hofer)f(and)g(E.)h(Zehnder,)h Fp(Symple)l(ctic)h(invariants)f (and)h(Hamiltonian)945 4385 y(dynamics)p FA(,)29 b(Birkh\177)-42 b(auser,)26 b(Basel,)h(1994.)515 4546 y([Kap91])147 b(T.)23 b(Kapp)r(eler,)f Fp(Fibr)l(ation)27 b(of)f(the)f(phase-sp)l(ac)l(e)i (for)f(the)g(Kortewe)l(g-de)g(Vries)945 4646 y(e)l(quation)p FA(,)i(Ann.)g(Inst.)g(F)-7 b(ourier)27 b FB(41)g FA(\(1991\),)f (539{575.)515 4807 y([Kat75])161 b(T.)32 b(Kato,)g Fp(Quasi-line)l(ar)i (e)l(quations)g(of)g(evolutions,)i(with)e(applic)l(ations)i(to)945 4907 y(p)l(artial)42 b(di\013er)l(ential)g(e)l(quations)p FA(,)h(Lecture)c(Notes)g(in)h(Mathematics,)i(v)n(ol.)945 5006 y(448,)26 b(pp.)i(25{70,)d(Springer,)i(Berlin,)g(1975.)1905 5255 y(45)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 46 45 bop 515 523 a FA([KM01])159 b(T.)38 b(Kapp)r(eler)f(and)i(M.)f (Mak)-5 b(aro)n(v,)39 b Fp(On)g(Birkho\013)i(c)l(o)l(or)l(dinates)g (for)f(KdV)p FA(,)945 623 y(Ann.)28 b(H.P)n(oincar)n(\023)-39 b(e)25 b FB(2)i FA(\(2001\),)g(807{856.)515 789 y([KP96])178 b(S.)30 b(B.)h(Kuksin)f(and)g(J.)g(P\177)-42 b(osc)n(hel,)30 b Fp(Invariant)j(Cantor)f(manifolds)j(of)e(quasi-)945 888 y(p)l(erio)l(dic)48 b(oscil)t(lations)h(for)f(a)e(nonline)l(ar)h (Schr\177)-42 b(odinger)49 b(e)l(quation)p FA(,)h(Ann.)945 988 y(Math.)28 b FB(143)e FA(\(1996\),)h(149{179.)515 1154 y([KP98])178 b(I.)34 b(M.)h(Kric)n(hev)n(er)c(and)j(D.)h(H.)g (Phong,)f Fp(Symple)l(ctic)j(forms)f(in)g(the)g(the)l(ory)945 1254 y(of)j(solitons)p FA(,)g(Surv.)e(Di\013er.)g(Geom.,)h(v)n(ol.)e (IV,)h(pp.)g(239{313,)f(In)n(t.)h(Press,)945 1353 y(Boston,)27 b(1998.)515 1519 y([KP03])178 b(T.)28 b(Kapp)r(eler)e(and)i(J.)f(P\177) -42 b(osc)n(hel,)26 b Fp(KAM)k(&)f(KdV)p FA(,)f(Springer,)f(2003.)515 1685 y([Kuk87])145 b(S.)23 b(B.)g(Kuksin,)h Fp(Hamiltonian)i(p)l (erturb)l(ations)g(of)g(in\014nite-dimensional)h(lin-)945 1785 y(e)l(ar)42 b(systems)g(with)h(an)f(imaginary)h(sp)l(e)l(ctrum)p FA(,)h(F)-7 b(unct.)42 b(Anal.)f(Appl.)h FB(21)945 1885 y FA(\(1987\),)26 b(192{205.)515 2051 y([Kuk88])145 b(S.)37 b(B.)h(Kuksin,)h Fp(Perturb)l(ations)g(of)h(quasip)l(erio)l(dic)h (solutions)e(of)h(in\014nite-)945 2150 y(dimensional)29 b(Hamiltonian)e(systems)p FA(,)e(Izv.)g(Ak)-5 b(ad.)24 b(Nauk)h(SSSR)g(Ser.)f(Mat.)945 2250 y FB(52)j FA(\(1988\),)f(41{63,)g (Engl.)h(T)-7 b(ransl.)26 b(in)i(Math.)g(USSR)g(Izv.)g FB(32:1)e FA(\(1989\).)515 2416 y([Kuk89])145 b(S.)22 b(B.)f(Kuksin,)h Fp(The)j(p)l(erturb)l(ation)f(the)l(ory)h(for)g(the)f (quasip)l(erio)l(dic)j(solutions)945 2516 y(of)j (in\014nite-dimensional)h(Hamiltonian)f(systems)g(and)g(its)f(applic)l (ations)j(to)945 2615 y(the)e(Kortewe)l(g)f({)h(de)g(Vries)g(e)l (quation)p FA(,)e(Math.)f(USSR)h(Sb)r(ornik)f FB(64)g FA(\(1989\),)945 2715 y(397{413.)515 2881 y([Kuk93])145 b(S.)42 b(B.)g(Kuksin,)i Fp(Ne)l(arly)g(inte)l(gr)l(able)f (in\014nite-dimensional)h(Hamiltonian)945 2980 y(systems)p FA(,)27 b(Springer-V)-7 b(erlag,)26 b(Berlin,)h(1993.)515 3147 y([Kuk94])145 b(S.)38 b(B.)f(Kuksin,)j Fp(KAM-the)l(ory)f(for)h(p) l(artial)g(di\013er)l(ential)g(e)l(quations)p FA(,)h(Pro-)945 3246 y(ceedings)d(of)g(the)h(First)g(Europ)r(ean)e(Congress)g(of)h (Mathematics,)k(v)n(ol.)37 b(2,)945 3346 y(Birkh\177)-42 b(auser,)26 b(1994,)f(pp.)j(123{157.)515 3512 y([Kuk95])145 b(S.)55 b(B.)f(Kuksin,)61 b Fp(In\014nite-dimensional)55 b(symple)l(ctic)h(c)l(ap)l(acities)g(and)g(a)945 3611 y(sque)l(ezing)40 b(the)l(or)l(em)h(for)g(Hamiltonian)g(PDEs)p FA(,)h(Comm.)d(Math.)g(Ph)n(ysics)945 3711 y FB(167)27 b FA(\(1995\),)f(531{552.)515 3877 y([Kuk98])145 b(S.)39 b(B.)f(Kuksin,)j Fp(A)f(KAM-the)l(or)l(em)f(for)i(e)l(quations)f(of)h (the)f(Kortewe)l(g{de)945 3977 y(Vries)30 b(typ)l(e)p FA(,)e(Rev.)f(Math.)h(&)f(Math.)h(Ph)n(ys.)f FB(10)g FA(\(1998\),)f(no.)i(3,)f(1{64.)515 4143 y([Kuk99])145 b(S.)25 b(B.)f(Kuksin,)g Fp(Sp)l(e)l(ctr)l(al)j(pr)l(op)l(erties)h(of)g (solutions)f(for)h(nonline)l(ar)f(PDEs)g(in)945 4242 y(the)32 b(turbulent)f(r)l(e)l(gime)p FA(,)g(Geometric)f(and)g(F)-7 b(unctional)30 b(Analysis)f FB(9)h FA(\(1999\),)945 4342 y(141{184.)515 4508 y([Kuk00])145 b(S.)39 b(B.)h(Kuksin,)h Fp(A)n(nalysis)g(of)h(Hamiltonian)f(PDEs)p FA(,)i(Oxford)38 b(Univ)n(ersit)n(y)945 4608 y(Press,)26 b(Oxford,)h(2000.)515 4774 y([Lax75])162 b(P)-7 b(.)28 b(D.)h(Lax,)f Fp(Perio)l(dic)33 b(solutons)d(of)h(the)g(KdV)g(e)l(quations)p FA(,)e(Comm)n(un.)f(Pure) 945 4873 y(Appl.)g(Math.)g FB(28)f FA(\(1975\),)f(141{188.)1905 5255 y(46)p eop PStoPSsaved restore %%Page: (46,47) 24 userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 -36.850394 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip /showpage{}def/copypage{}def/erasepage{}def PStoPSxform concat 47 46 bop 515 523 a FA([Lo)r(c92])167 b(P)-7 b(.)29 b(Lo)r(c)n(hak,)g Fp(Canonic)l(al)k(p)l(erturb)l(ation)f(the)l(ory)g(via)h(simultane)l (ous)e(appr)l(ox-)945 623 y(imation)p FA(,)e(Russ.)e(Math.)h(Surv.)f FB(47)g FA(\(1992\),)g(no.)g(6,)g(57{133.)515 789 y([LS88])202 b(B.)28 b(V.)h(Lidskij)f(and)h(E.)f(I.)g(Sh)n(ulman,)h Fp(Perio)l(dic)j(solutions)f(of)g(the)g(e)l(quation)945 888 y Fz(u)993 900 y Fw(tt)1065 888 y Fy(\000)18 b Fz(u)1196 900 y Fw(xx)1294 888 y FA(+)g Fz(u)1425 858 y Fv(3)1484 888 y FA(=)23 b(0,)k(F)-7 b(unct.)29 b(Anal.)e(Appl.)i FB(22)e FA(\(1988\),)f(332{333.)515 1054 y([Mos76])149 b(J.)22 b(Moser,)g Fp(Perio)l(dic)27 b(orbits)f(ne)l(ar)e(an)h(e)l (quilibrium)h(and)f(a)h(the)l(or)l(em)f(by)g(Alan)945 1154 y(Weinstein)p FA(,)j(Comm.)f(Pure)g(Appl.)h(Math.)g FB(29)f FA(\(1976\),)g(724{747.)515 1320 y([MS71])178 b(J.)23 b(Moser)f(and)g(C.)h(L.)g(Siegel,)h Fp(L)l(e)l(ctur)l(es)h(on)g (c)l(elestial)i(me)l(chanics)p FA(,)e(Springer,)945 1420 y(Berlin,)i(1971.)515 1586 y([Nek77])157 b(N.)25 b(N.)g(Nekhoroshev,)f Fp(Exp)l(onential)k(estimate)f(of)h(the)g(stability)g(of)g(ne)l(ar)f (in-)945 1685 y(te)l(gr)l(able)d(Hamiltonian)g(systems)p FA(,)e(Russ.)f(Math.)g(Surv)n(eys)f FB(32)g FA(\(1977\),)h(no.)f(6,)945 1785 y(1{65.)515 1951 y([Nie98])178 b(L.)45 b(Niederman,)j Fp(Nonline)l(ar)e(stability)h(ar)l(ound)f(an)g(el)t(liptic)h(e)l (quilibrium)945 2051 y(p)l(oint)24 b(in)f(an)h(Hamiltonian)h(system)p FA(,)d(Nonlinearit)n(y)f FB(11)f FA(\(1998\),)h(1465{1479.)515 2217 y([No)n(v74])154 b(S.)30 b(P)-7 b(.)30 b(No)n(vik)n(o)n(v,)f Fp(A)j(p)l(erio)l(dic)i(pr)l(oblem)f(for)g(the)f(Kortewe)l(g-de-Vries)h (e)l(qua-)945 2316 y(tion,)d(I)p FA(,)e(F)-7 b(unct.)28 b(Anal.)g(Appl.)g FB(8)g FA(\(1974\),)e(236{246.)515 2482 y([P)n(az83])166 b(A.)30 b(P)n(azy)-7 b(,)29 b Fp(Semigr)l(oups)j (of)h(line)l(ar)f(op)l(er)l(ators)h(and)f(applic)l(ations)i(to)e(p)l (artial)945 2582 y(di\013er)l(ential)f(e)l(quations)p FA(,)d(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1983.)515 2748 y([P\177)-42 b(os89])168 b(J.)30 b(P\177)-42 b(osc)n(hel,)29 b Fp(On)i(el)t(liptic)i(lower)g(dimensional)h(tori)e(in)g(Hamiltoniam)h (sys-)945 2848 y(tems)p FA(,)27 b(Math.)h(Z.)f FB(202)g FA(\(1989\),)g(559{608.)515 3014 y([P\177)-42 b(os90])168 b(J.)34 b(P\177)-42 b(osc)n(hel,)35 b Fp(smal)t(l)i(divisors)i(with)d (sp)l(atial)i(structur)l(e)c(in)j(in\014nite)f(dimen-)945 3113 y(sional)31 b(Hamiltonian)g(systems)p FA(,)d(Comm.)g(Math.)g(Ph)n (ys.)f FB(127)g FA(\(1990\),)g(351{)945 3213 y(393.)515 3379 y([P\177)-42 b(os96a])126 b(J.)23 b(P\177)-42 b(osc)n(hel,)24 b Fp(A)i(KAM-the)l(or)l(em)g(for)h(some)g(nonline)l(ar)f(PDEs)p FA(,)f(Ann.)g(Scuola)945 3479 y(Norm.)i(Sup.)h(Pisa,)f(Cl.)g(Sci.,)h (IV)g(Ser.)g(15)e FB(23)h FA(\(1996\),)g(119{148.)515 3645 y([P\177)-42 b(os96b])122 b(J.)29 b(P\177)-42 b(osc)n(hel,)29 b Fp(Quasi-p)l(erio)l(dic)34 b(solutions)e(for)h(a)f(nonline)l(ar)g (wave)h(e)l(quation)p FA(,)945 3744 y(Commen)n(t.)27 b(Math.)h(Helv.)g FB(71)f FA(\(1996\),)f(269{296.)515 3910 y([P\177)-42 b(os99])168 b(J.)31 b(P\177)-42 b(osc)n(hel,)31 b Fp(On)h(Nekhor)l(oshev)k(estimates)d(for)h(a)g(nonline)l(ar)f (Schr\177)-42 b(odinger)945 4010 y(e)l(quation)29 b(and)h(a)g(the)l(or) l(em)f(by)h(Bambusi)p FA(,)e(Nonlinearit)n(y)g FB(12)e FA(\(1999\),)g(1587{)945 4110 y(1600.)515 4276 y([PT01])183 b(P)-7 b(.)30 b(Plotnik)n(o)n(v)e(and)i(J.)g(T)-7 b(oland,)30 b Fp(Nash-Moser)j(the)l(ory)g(for)g(standing)g(water)945 4375 y(waves)p FA(,)28 b(Arc)n(h.)g(Ration.)f(Mec)n(h.)h(Anal.)f FB(159)g FA(\(2001\),)f(1{83.)515 4541 y([Rab78])151 b(P)-7 b(.)21 b(Rabino)n(witz,)i Fp(F)-6 b(r)l(e)l(e)24 b(vibr)l(ations)i(for)f(a)g(semiline)l(ar)h(wave)f(e)l(quation)p FA(,)f(Com-)945 4641 y(m)n(un.)k(Pure)e(Appl.)j(Math.)e FB(31)h FA(\(1978\),)e(31{68.)515 4807 y([RS75])193 b(M.)32 b(Reed)g(and)g(B.)g(Simon,)h Fp(Metho)l(ds)i(of)g(mo)l(dern)f (mathematic)l(al)h(physics)p FA(,)945 4907 y(v)n(ol.)27 b(2,)g(Academic)g(Press,)g(New)g(Y)-7 b(ork)27 b(-)h(London,)f(1975.) 1905 5255 y(47)p eop PStoPSsaved restore userdict/PStoPSsaved save put PStoPSmatrix setmatrix 680.314961 340.157480 translate 90 rotate 0.900000 dup scale userdict/PStoPSmatrix matrix currentmatrix put userdict/PStoPSclip{0 0 moveto 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto closepath}put initclip PStoPSxform concat 48 47 bop 515 523 a FA([Sev03])173 b(M.)21 b(B.)g(Sevryuk,)g Fp(The)k(classic)l(al)g(KAM)f(the)l(ory)g(at)g(the)f(dawn)i(of)f(the)g (twenty-)945 623 y(\014rst)29 b(c)l(entury)p FA(,)e(Mosco)n(w)f(Math.)i (J.)f FB(3)g FA(\(2003\).)515 789 y([W)-7 b(a)n(y84])138 b(C.)32 b(E.)g(W)-7 b(a)n(yne,)32 b Fp(The)j(KAM)f(the)l(ory)g(of)h (systems)f(with)g(short)g(r)l(ange)g(inter-)945 888 y(action,)46 b(1)c(and)h(2)p FA(,)h(Comm.)d(Math.)g(Ph)n(ysics)f FB(96)g FA(\(1984\),)j(311{329)37 b(and)945 988 y(331{344.)515 1154 y([W)-7 b(a)n(y90])138 b(C.)43 b(E.)f(W)-7 b(a)n(yne,)46 b Fp(Perio)l(dic)g(and)f(quasi-p)l(erio)l(dic)h(solutions)e(of)g (nonline)l(ar)945 1254 y(wave)27 b(e)l(quations)g(via)g(KAM)f(the)l (ory)p FA(,)g(Comm.)e(Math.)g(Ph)n(ysics)e FB(127)h FA(\(1990\),)945 1353 y(479{528.)515 1519 y([W)-7 b(ei73])162 b(A.)40 b(W)-7 b(einstein,)43 b Fp(Normal)f(mo)l(des)f(for)h(nonline)l(ar)f (Hamiltonian)h(systems)p FA(,)945 1619 y(In)n(v)n(en)n(t.)27 b(Math.)h FB(20)f FA(\(1973\),)f(47{57.)515 1785 y([Zak68])163 b(V.)35 b(E.)g(Zakharo)n(v,)f Fp(Stability)j(of)g(p)l(erio)l(dic)i (waves)e(of)g(\014nite)f(amplitude)i(on)945 1885 y(the)i(surfac)l(e)h (of)g(a)g(de)l(ep)g(\015uid)p FA(,)h(Appl.)e(Mec)n(h.)f(T)-7 b(ec)n(h.)39 b(Ph)n(ysics)e FB(2)i FA(\(1968\),)945 1984 y(190{194.)515 2150 y([Zhi01])180 b(P)-7 b(.)26 b(E.)g(Zhidk)n(o)n(v,)f Fp(Kortewe)l(g-de)30 b(Vries)f(and)g(Nonline)l(ar)g(Schr\177)-42 b(odinger)30 b(e)l(qua-)945 2250 y(tions:)39 b(qualitative)31 b(the)l(ory)p FA(,)d(Springer-V)-7 b(erlag,)25 b(Berlin,)i(2001.)515 2416 y([ZIS79])173 b(V.)30 b(E.)f(Zakharo)n(v,)f(M.)h(F.)h(Iv)-5 b(ano)n(v,)29 b(and)h(L.)f(N.)h(Sh)n(ur,)g Fp(On)h(the)h(abnormal)t(ly) 945 2516 y(slow)40 b(sto)l(chastisation)h(in)e(some)h(two-dimensional)h (\014eld)f(the)l(ory)g(mo)l(dels)p FA(,)945 2615 y(JETP)26 b(Letters)h FB(30)g FA(\(1979\),)g(no.)g(1,)g(39{44.)515 2781 y([ZMNP84])54 b(V.)23 b(E.)g(Zakharo)n(v,)e(S.)i(V.)g(Manak)n(o)n (v,)f(S.)h(P)-7 b(.)23 b(No)n(vik)n(o)n(v,)f(and)g(L.)h(P)-7 b(.)23 b(Pitaevskij,)945 2881 y Fp(The)l(ory)31 b(of)g(solitons)p FA(,)d(Plen)n(um)f(Press,)g(New)g(Y)-7 b(ork,)27 b(1984.)639 3146 y FB(Sergei)k(B.)g(Kuksin)639 3245 y Fp(Dep)l(artment)e(of)i (Mathematics)639 3345 y(Heriot-Watt)f(University)639 3444 y(Edinbur)l(gh)h(EH14)g(4AS)639 3544 y(Sc)l(otland,)g(UK)724 3644 y FA(and)639 3743 y Fp(Steklov)f(Institute)f(of)h(Mathematics)639 3843 y(8)g(Gubkina)h(St.,)f(117966)i(Mosc)l(ow)639 3943 y(R)n(ussia)639 4067 y(E-mail:)40 b Fa(kuksin@ma.hw.ac.)o(uk)639 4192 y Fp(web-addr)l(ess)p FA(:)f Fa(www.ma.hw.ac.uk)o(/)p Fy(\030)p Fa(k)o(uk)o(sin)o(/)639 4364 y FB(Dario)32 b(Bam)m(busi)639 4464 y Fp(Dip)l(artimento)f(di)f(Mathematic)l(a)639 4563 y(Via)h(Saldini)g(50,)g(20133)h(Milano,)f(Italy)639 4688 y(E-mail:)40 b Fa(dario.bambusi@un)o(im)o(i.i)o(t)639 4813 y Fp(web-addr)l(ess)p FA(:)f Fa(http://users.ma)o(t.u)o(ni)o(mi.)o (it)o(/u)o(ser)o(s/)o(bam)o(bu)o(si)o(/)1905 5255 y FA(48)p eop PStoPSsaved restore %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306030901183--