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3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fs 539 1022 a(BIRKHOFF) k(NORMAL) g(F) m(ORM) h(F) m(OR) f (SOME) g(QUASILINEAR) 1320 1130 y(HAMIL) -8 b(TONIAN) 32 b(PDES) p Fr 1498 1471 a(D) n(ARIO) 24 b(BAMBUSI) p Fq 711 1587 a(Dip) l(artimento) k(di) f(Matematic) l(a,) i(Via) e(Saldini) f(50,) i(20) g(133) g(Milano,) e(Italy) p Fp 562 1934 a(Consider) h(a) f(Hamiltonian) f(PDE) h(ha) n(ving) h(an) g(elliptic) f (equilibrium) e(at) j(zero.) 39 b(Assuming) 25 b(a) 562 2017 y(suitable) f(condition) g(on) f(the) h(frequencies) g(w) n(e) f (will) e(construct) k(a) e(canonical) h(transformation) 562 2100 y(putting) 29 b(the) f(system) f(in) g(Birkho\013) g(normal) f (form) g(up) h(to) h(a) g(small) d(reminder.) 41 b(In) 27 b(the) i(case) 562 2183 y(of) f(quasilinear) g(systems) f(the) h (normal) f(form) f(will) g(b) r(e) j(used) f(to) g(describ) r(e) h(the) f(dynamics) g(of) 562 2266 y(smo) r(oth) 22 b(solutions) f(of) g(small) f(amplitude.) 30 b(Applications) 22 b(to) g(the) g(w) n(ater) g(w) n(a) n(v) n(e) g(problem) f(and) 562 2349 y(to) k(quasilinear) e(w) n(a) n (v) n(e) h(equations) h(will) e(b) r(e) h(giv) n(en.) p Fs 172 2765 a(1.) 47 b(In) m(tro) s(duction) p Ft 172 2914 a(During) 33 b(the) g(last) f(\014fteen) i(y) n(ears) d(p) r (erturbation) h(theory) g(of) h(Hamiltonian) f(partial) g(di\013eren) n (tial) g(equa-) 172 3022 y(tions) g(has) e(b) r(een) i(quite) g (extensiv) n(ely) f(studied) g(and) h(remark) -5 b(able) 30 b(results) g(ha) n(v) n(e) h(b) r(een) g(established.) 48 b(In) 172 3130 y(particular) 38 b(the) i(existence) f(of) g(quasip) r (erio) r(dic) f(solutions) h(has) f(b) r(een) i(pro) n(v) n(ed) e (through) g(suitable) h(ex-) 172 3238 y(tensions) 32 b(of) f(KAM) h(theory) p Fo 1063 3208 a(18) p Fn(;) p Fo(20) p Fn(;) p Fo(17) p Fn(;) p Fo(11) p Fn(;) p Fo(6) p Fn(;) p Fo(8) p Ft 1496 3238 a(.) 48 b(Ho) n(w) n(ev) n(er) 30 b(v) n(ery) g(little) j(is) e(kno) n(wn) g(on) g(the) h(b) r(eha) n (vior) f(of) g(the) 172 3346 y(solutions) 23 b(lying) h(outside) f(KAM) h(tori,) g(solutions) f(whic) n(h) g(corresp) r(ond) f(to) i(the) g (large) e(ma) 5 b(jorit) n(y) 23 b(of) g(initial) 172 3454 y(data.) 51 b(In) 33 b(the) f(\014nite) h(dimensional) f(case) g (they) g(are) f(describ) r(ed) i(b) n(y) f(Nekhoroshev's) e(theorem,) j (whose) 172 3562 y(extension) 26 b(to) h(PDEs) f(is) g(at) g(presen) n (t) g(a) g(completely) h(op) r(en) f(problem) p Fo 2325 3531 a(a) p Ft 2362 3562 a(.) 37 b(Ho) n(w) n(ev) n(er) 24 b(there) i(is) h(a) f(particular) 172 3669 y(situation) i(where) f(the) h(full) h(p) r(o) n(w) n(er) d(of) i(Nekhoroshev) e(theorem) h(is) h (not) g(needed:) 37 b(the) 28 b(neigh) n(b) r(orho) r(o) r(d) f(of) 172 3777 y(an) 33 b(elliptic) h(equilibrium) g(p) r(oin) n(t,) h(where) e (\(in) h(the) f(\014nite) h(dimensional) f(case\)) g(Birkho\013) g (normal) f(form) 172 3885 y(theorem) c(giv) n(es) e(a) h(quite) h (precise) f(description) g(of) g(the) h(dynamics.) 297 3993 y(In) 19 b(the) h(presen) n(t) 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n(ell) f(kno) n(wn) g(that) h(there) f(exist) h (co) r(ordinates) e(in) i(whic) n(h) f(the) h(Hamiltonian) g(tak) n(es) e(the) i(form) p Fm 1593 1292 a(H) p Ft 30 w(=) p Fm 22 w(H) p Fo 1848 1304 a(0) p Ft 1904 1292 a(+) p Fm 18 w(P) p Ft 1326 w(\(1\)) 172 1475 y(where) p Fm 1362 1710 a(H) p Fo 1431 1722 a(0) p Ft 1468 1710 a(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) 23 b(:=) p Fn 1816 1606 a(N) p Fh 1785 1631 a(X) p Fn 1788 1808 a(j) p Fo 3 w(=1) p Fm 1919 1710 a(!) p Fn 1971 1722 a(j) p Fm 2015 1645 a(p) p Fo 2057 1615 a(2) p Fn 2057 1666 a(j) p Ft 2113 1645 a(+) p Fm 18 w(q) p Fo 2236 1615 a(2) p Fn 2233 1666 a(j) p 2015 1691 258 4 v Ft 2124 1767 a(2) 3366 1710 y(\(2\)) 172 1968 y(and) p Fm 29 w(P) p Ft 40 w(has) 28 b(a) g(zero) g(of) g(order) f(at) i(least) f(three) g(at) h(the) g (origin.) 38 b(Assume) 29 b(that) p Fm 29 w(P) p Ft 40 w(is) f(of) h(class) p Fm 27 w(C) p Fk 3188 1938 a(1) p Ft 3259 1968 a(,) g(then) 172 2076 y(the) f(follo) n(wing) f(theorem) g (holds.) p Fs 172 2270 a(Theorem) 42 b(2.1.) p Fg 41 w([Birkho\013) 9 b(]) 40 b(F) -6 b(or) 39 b(any) g(p) l(ositive) h (inte) l(ger) p Fm 38 w(M) p Fl 47 w(\025) p Ft 38 w(2) p Fg 38 w(ther) l(e) f(exist) f(a) h(neighb) l(orho) l(o) l(d) i(of) 172 2378 y(the) 34 b(origin) p Fl 35 w(U) p Fn 613 2390 a(M) p Fg 721 2378 a(and) g(a) g(c) l(anonic) l(al) h(tr) l(ansformation) p Fl 34 w(T) p Fn 1946 2390 a(M) p Ft 2051 2378 a(:) p Fl 30 w(U) p Fn 2156 2390 a(M) p Fl 2260 2378 a(!) p Ff 30 w(R) p Fo 2427 2347 a(2) p Fn(N) p Fg 2563 2378 a(which) h(puts) d(the) h(system) f(in) 172 2485 y(Birkho\013) f (normal) e(form,) h(namely) g(such) f(that) p Fm 1375 2668 a(H) p Fl 25 w(\016) p Fm 18 w(T) p Fn 1578 2680 a(M) p Ft 1674 2668 a(=) p Fm 23 w(H) p Fo 1831 2680 a(0) p Ft 1886 2668 a(+) p Fm 18 w(Z) p Ft 24 w(+) p Fm 19 w(R) p Fn 2197 2680 a(M) p Ft 3366 2668 a(\(3\)) p Fg 172 2851 a(wher) l(e) p Fm 31 w(Z) p Fg 35 w(Poisson) h(c) l (ommutes) e(with) p Fm 31 w(H) p Fo 1446 2863 a(0) p Fg 1483 2851 a(,) h(namely) p Fl 30 w(f) p Fm(H) p Fo 1935 2863 a(0) p Ft 1972 2851 a(;) p Fm 14 w(Z) p Fl 6 w(g) 22 b(\021) p Ft 23 w(0) p Fg 29 w(and) p Fm 30 w(R) p Fn 2519 2863 a(M) p Fg 2623 2851 a(is) 30 b(smal) t(l,) h(i.e.) p Fl 1256 3034 a(k) p Fm(X) p Fn 1367 3046 a(R) p Fe 1417 3054 a(M) p Ft 1482 3034 a(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) p Fl(k) 23 b(\024) p Fm 23 w(C) p Fn 1877 3046 a(M) p Fl 1965 3034 a(k) p Ft -1 w(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) p Fl(k) p Fn 2231 2992 a(M) p Fo 6 w(+1) p Ft 3366 3034 a(\(4\)) p Fg 172 3217 a(wher) l(e) p Fm 31 w(X) p Fn 476 3229 a(R) p Fe 526 3237 a(M) p Fg 622 3217 a(is) 30 b(the) g(Hamiltonian) g(ve) l(ctor) g(\014eld) g(of) p Fm 31 w(R) p Fn 1912 3229 a(M) p Fm 1357 3441 a(X) p Fn 1426 3453 a(R) p Fe 1476 3461 a(M) p Ft 1565 3441 a(=) p Fh 1653 3324 a(\022) p Fl 1714 3441 a(\000) p Fm 1789 3385 a(@) 5 b(R) p Fn 1901 3397 a(M) p 1789 3422 186 4 v Fm 1821 3498 a(@) g(q) p Fn 1907 3510 a(j) p Fm 1984 3441 a(;) 2031 3385 y(@) g(R) p Fn 2143 3397 a(M) p 2031 3422 V Fm 2061 3498 a(@) g(p) p Fn 2152 3510 a(j) p Fh 2226 3324 a(\023) p Fs 172 3673 a(Remark) 31 b(2.1.) p Ft 40 w(In) d(the) g(case) f(where) g(the) h(frequencies) f (are) g(nonresonan) n(t) e(namely) j(they) f(ful\014ll) p Fm 1337 3856 a(!) p Fl 21 w(\001) p Fm 19 w(k) p Fl 25 w(6) p Ft(=) c(0) p Fm 27 w(;) p Fl 97 w(8) p Fm(k) p Fl 25 w(2) p Ff 24 w(Z) p Fn 2052 3822 a(n) p Fl 2091 3856 a(nf) p Ft(0) p Fl(g) p Fm 26 w(;) p Ft 1058 w(\(5\)) 172 4039 y(the) 28 b(function) p Fm 29 w(Z) p Ft 33 w(can) f(b) r(e) h(sho) n(wn) f(to) g(dep) r(end) i(on) e(the) h(actions) p Fm 27 w(I) p Fn 2214 4051 a(j) p Ft 2272 4039 a(:=) 23 b(\() p Fm(p) p Fo 2457 4009 a(2) p Fn 2457 4061 a(j) p Ft 2513 4039 a(+) p Fm 18 w(q) p Fo 2636 4009 a(2) p Fn 2633 4061 a(j) p Ft 2673 4039 a(\)) p Fm(=) p Ft(2) k(only) -7 b(.) 297 4232 y(In) 28 b(the) g(nonresonan) n(t) e(case) g(one) i(can) f (immediately) g(deduce) h(some) f(dynamical) g(consequences:) p Fs 172 4426 a(Corollary) 35 b(2.1.) p Fg 41 w(Assume) d(that) g(c) l (ondition) h(\(5\)) g(holds,) i(then) d(for) h(any) p Fm 32 w(M) p Fo 2589 4438 a(1) p Fm 2626 4426 a(;) 14 b(M) p Fo 2744 4438 a(2) p Fg 2813 4426 a(with) p Fm 33 w(M) p Ft 36 w(:=) 27 b(2) p Fm(M) p Fo 3351 4438 a(1) p Ft 3408 4426 a(+) p Fm 172 4533 a(M) p Fo 253 4545 a(2) p Fl 313 4533 a(\025) p Ft 23 w(2) p Fg 29 w(ther) l(e) j(exists) p Fm 29 w(\017) p Fn 940 4545 a(M) p Fg 1043 4533 a(with) h(the) f(fol) t(lowing) i(pr) l(op) l (erty.) 39 b(Consider) 32 b(a) e(solution) p Fm 30 w(z) p Ft 4 w(\() p Fm(t) p Ft(\)) p Fg 29 w(c) l(orr) l(esp) l(onding) 172 4641 y(to) g(an) g(initial) h(datum) p Fm 30 w(z) p Fo 929 4653 a(0) p Fl 988 4641 a(\021) p Ft 23 w(\() p Fm(p) p Fo 1150 4611 a(0) p Fm 1187 4641 a(;) 14 b(q) p Fo 1264 4611 a(0) p Ft 1302 4641 a(\)) p Fg 30 w(ful\014l) t(ling) p Fm 1524 4824 a(\017) p Ft 23 w(:=) p Fl 22 w(k) p Fm(z) p Fo 1772 4836 a(0) p Fl 1809 4824 a(k) p Fm 22 w(<) 23 b(\017) p Fn 1995 4836 a(M) p Fm 2098 4824 a(:) p Fg 172 5007 a(Then) 31 b(ther) l(e) f(exists) f(a) h(smo) l(oth) g(torus) p Ff 30 w(T) p Fn 1447 5019 a(z) p Fd 1479 5027 a(0) p Fg 1544 5007 a(such) g(that) p Fm 1010 5190 a(d) p Ft(\() p Fm(z) p Ft 4 w(\() p Fm(t) p Ft(\)) p Fm(;) p Ff 14 w(T) p Fn 1315 5202 a(z) p Fd 1347 5210 a(0) p Ft 1383 5190 a(\)) p Fl 23 w(\024) p Fm 23 w(C) p Fn 1585 5202 a(M) p Fm 1659 5190 a(\017) p Fn 1693 5156 a(M) p Fd 1756 5164 a(1) p Fg 1877 5190 a(for) p Fl 86 w(j) p Fm(t) p Fl(j) 23 b(\024) p Ft 23 w(\() p Fm(C) p Fn 2343 5202 a(M) p Fm 2417 5190 a(\017) p Ft(\)) p Fk 2483 5156 a(\000) p Fn(M) p Fd 2598 5164 a(2) p Ft 3366 5190 a(\(6\)) p Fg 172 5373 a(Mor) l(e) l(over,) 32 b(up) e(to) f(the) h(same) g(time) g(one) g (has) p Fl 1132 5556 a(k) p Fm(z) p Ft 4 w(\() p Fm(t) p Ft(\)) p Fl(k) 22 b(\024) p Ft 23 w(2) p Fm(\017) 29 b(;) p Fl 183 w(j) p Fm(I) p Fn 1833 5568 a(j) p Ft 1869 5556 a(\() p Fm(t) p Ft(\)) p Fl 19 w(\000) p Fm 18 w(I) p Fn 2101 5568 a(j) p Ft 2136 5556 a(\(0\)) p Fl(j) 24 b(\024) p Fm 22 w(C) 6 b(\017) p Fo 2475 5522 a(3) p Ft 3366 5556 a(\(7\)) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 3437 636 a(3) p Ft 297 852 a(The) 34 b(main) h(di\016cult) n (y) f(in) h(order) e(to) h(extend) h(theorem) f(2.1) f(to) h(the) h (in\014nite) g(dimensional) f(case) g(is) 172 960 y(due) c(to) g(the) g (app) r(earance) e(of) i(small) f(denominators.) 42 b(In) 30 b(the) g(\014nite) h(dimensional) e(case) g(one) g(has) g(that,) 172 1068 y(ha) n(ving) e(\014xed) p Fm 28 w(M) p Ft 9 w(,) 892 1228 y([0) p Fm 23 w(<) p Fl 23 w(j) p Fm(k) p Fl 3 w(j) c(\024) p Fm 22 w(M) p Ft 27 w(+) 18 b(2) p Fm 27 w(;) 97 b(!) p Fl 21 w(\001) p Fm 19 w(k) p Fl 26 w(6) p Ft(=) 22 b(0]) 42 b(implies) p Fl 27 w(j) p Fm(!) p Fl 22 w(\001) p Fm 18 w(k) p Fl 3 w(j) 23 b(\025) p Fm 23 w(\015) 32 b(;) p Ft 641 w(\(8\)) 172 1388 y(a) 39 b(condition) f(needed) h(in) g(the) g (pro) r(of) f(of) g(Birkho\013) 6 b('s) 39 b(theorem.) 69 b(In) 39 b(the) g(in\014nite) g(dimensional) f(case) 172 1496 y(\() p Fm(N) p Ft 33 w(=) p Fl 22 w(1) p Ft(\)) 28 b(the) g(set) g(considered) e(in) i(\(8\)) g(is) f(in\014nite) i(and) e (in) h(all) f(in) n(teresting) g(cases) g(one) g(has) 1560 1657 y(inf) p Fd 1452 1702 a(0) p Fe(<) p Fc(j) p Fe(k) p Fc 1 w(j\024) p Fe(M) p Fd 5 w(+2) p Fe 1531 1749 a(!) p Fc 1 w(\001) p Fe(k) p Fc 1 w(6) p Fd(=0) p Fl 1793 1657 a(j) p Fm(!) p Fl 21 w(\001) p Fm 19 w(k) p Fl 3 w(j) p Ft 23 w(=) 22 b(0) p Fm 28 w(:) p Ft 172 1884 a(This) 28 b(mak) n(es) f(imp) r(ossible) g(a) g(straigh) n(tforw) n(ard) e (generalization) g(of) j(Birkho\013) 6 b('s) 27 b(theorem) g(to) h (PDEs.) p Fs 172 2143 a(3.) 47 b(An) 32 b(abstract) i(result) d(for) h (the) g(in\014nite) f(dimensional) d(case) p Ft 172 2293 a(Consider) f(a) g(\(formal\)) h(Hamiltonian) f(system) g(of) h(the) g (form) f(\(1\)) h(with) p Fm 1362 2507 a(H) p Fo 1431 2519 a(0) p Ft 1468 2507 a(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) 23 b(:=) p Fh 1785 2428 a(X) p Fn 1787 2606 a(j) p Fk 3 w(\025) p Fo(1) p Fm 1919 2507 a(!) p Fn 1971 2519 a(j) p Fm 2015 2442 a(p) p Fo 2057 2412 a(2) p Fn 2057 2463 a(j) p Ft 2113 2442 a(+) p Fm 18 w(q) p Fo 2236 2412 a(2) p Fn 2233 2463 a(j) p 2015 2488 258 4 v Ft 2124 2564 a(2) 3366 2507 y(\(9\)) 172 2754 y(T) -7 b(o) 29 b(de\014ne) g(precisely) f(the) h(phase) g(space) f(consider) g(the) h (Banac) n(h) f(space) p Fm 28 w(`) p Fo 2503 2724 a(2) p Fn 2503 2775 a(s) p Ft 2569 2754 a(of) h(the) g(sequences) p Fl 28 w(f) p Fm(x) p Fn 3276 2766 a(j) p Fl 3311 2754 a(g) p Fn 3353 2766 a(j) p Fk 3 w(\025) p Fo(1) p Ft 172 2862 a(suc) n(h) f(that) p Fl 1386 3023 a(k) p Fm(x) p Fl(k) p Fo 1517 2981 a(2) p Fn 1517 3047 a(s) p Ft 1577 3023 a(:=) p Fh 1688 2944 a(X) p Fn 1690 3122 a(j) p Fk 3 w(\025) p Fo(1) p Fm 1822 3023 a(j) p Fo 1861 2988 a(2) p Fn(s) p Fm 1929 3023 a(x) p Fo 1976 2988 a(2) p Fn 1976 3043 a(j) p Fm 2037 3023 a(<) p Fl 23 w(1) p Fm 27 w(;) p Ft 172 3281 a(and) 33 b(denote) p Fl 32 w(P) p Fn 669 3293 a(s) p Ft 736 3281 a(:=) p Fm 31 w(`) p Fo 890 3251 a(2) p Fn 890 3302 a(s) p Fl 948 3281 a(\002) p Fm 21 w(`) p Fo 1069 3251 a(2) p Fn 1069 3302 a(s) p Ft 1106 3281 a(.) 52 b(If) p Fm 33 w(z) p Fl 34 w(\021) p Ft 31 w(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) p Fl 32 w(2) 31 b(P) p Fn 1797 3293 a(s) p Ft 1865 3281 a(is) h(a) h(phase) f(p) r(oin) n(t,) i(w) n(e) e(will) g(denote) h(b) n(y) p Fl 32 w(k) p Fm(z) p Fl 4 w(k) p Fo 3316 3240 a(2) p Fn 3316 3306 a(s) p Ft 3385 3281 a(:=) p Fl 172 3389 a(k) p Fm(p) p Fl(k) p Fo 297 3348 a(2) p Fn 297 3414 a(s) p Ft 353 3389 a(+) p Fl 18 w(k) p Fm -1 w(q) p Fl 3 w(k) p Fo 559 3348 a(2) p Fn 559 3414 a(s) p Ft 624 3389 a(\(the) 28 b(square) e(of) 6 b(\)) 29 b(its) f(norm) f(and) g(b) n(y) p Fm 27 w(B) p Fn 1866 3401 a(s) p Ft 1902 3389 a(\() p Fm(R) p Ft 1 w(\)) h(the) g(op) r(en) f(ball) h(of) f(radius) p Fm 27 w(R) p Ft 28 w(in) p Fl 28 w(P) p Fn 3153 3401 a(s) p Ft 3188 3389 a(.) 297 3497 y(F) -7 b(or) 27 b(an) n(y) g(p) r (ositiv) n(e) g(\(large\)) p Fm 26 w(N) p Ft 37 w(denote) g(b) n(y) p Fm 27 w(!) p Fo 1718 3467 a(\() p Fn(N) p Fo 6 w(\)) p Ft 1856 3497 a(:=) c(\() p Fm(!) p Fo 2051 3509 a(1) p Fm 2088 3497 a(;) 14 b(:::;) g(!) p Fn 2283 3509 a(N) p Ft 2345 3497 a(\)) 28 b(the) g(truncation) f(of) h(length) p Fm 27 w(N) p Ft 37 w(of) 172 3605 y(the) g(frequency) g(v) n(ector.) 35 b(W) -7 b(e) 28 b(assume) f(that) 294 3769 y(H1\)) 41 b(F) -7 b(or) 28 b(an) n(y) g(p) r(ositiv) n(e) g(\(large\)) p Fm 27 w(r) p Ft 32 w(there) g(exist) p Fm 28 w(\013) p Ft 25 w(=) p Fm 24 w(\013) p Ft(\() p Fm(r) p Ft 2 w(\)) j(and) p Fm 28 w(\015) p Ft 29 w(=) p Fm 24 w(\015) p Ft 5 w(\() p Fm(r) p Ft 2 w(\)) p Fm 25 w(>) p Ft 24 w(0) e(suc) n(h) f(that) g(for) g(an) n(y) p Fm 471 3877 a(N) p Ft 37 w(and) p Fl 27 w(8) p Fm(k) p Fl 25 w(2) p Ff 24 w(Z) p Fn 991 3847 a(N) p Ft 1048 3877 a(,) g(with) g(0) p Fm 22 w(<) p Fl 23 w(j) p Fm(k) p Fl 3 w(j) 23 b(\024) p Fm 23 w(r) p Ft 21 w(+) 18 b(2) 27 b(one) g(has) 1166 4050 y(either) p Fm 83 w(!) p Fo 1512 4016 a(\() p Fn(N) p Fo 6 w(\)) p Fl 1645 4050 a(\001) p Fm 18 w(k) p Ft 26 w(=) c(0) 82 b(or) p Fl 82 w(j) p Fm(!) p Fo 2202 4016 a(\() p Fn(N) p Fo 6 w(\)) p Fl 2335 4050 a(\001) p Fm 19 w(k) p Fl 3 w(j) 23 b(\025) p Fm 2604 3994 a(\015) p 2566 4031 124 4 v 2566 4107 a(N) p Fn 2642 4083 a(\013) p Fm 2727 4050 a(;) p Ft 471 4230 a(where) p Fl 27 w(j) p Fm(k) p Fl 3 w(j) p Ft 23 w(:=) p Fl 23 w(j) p Fm(k) p Fo 1003 4242 a(1) p Fl 1041 4230 a(j) p Ft 18 w(+) p Fm 18 w(:::) p Ft 19 w(+) p Fl 18 w(j) p Fm(k) p Fn 1402 4242 a(N) p Fl 1465 4230 a(j) p Ft(.) 294 4338 y(H2\)) 41 b(There) 23 b(exists) p Fm 23 w(s) p Fo 971 4350 a(0) p Ft 1009 4338 a(,) h(and,) g(for) f(an) n(y) g(p) r(ositiv) n(e) p Fm 23 w(r) p Ft 26 w(there) g(exists) p Fm 24 w(d) p Fn 2356 4350 a(r) p Ft 2416 4338 a(with) h(the) g(follo) n(wing) e (prop) r(erties:) 471 4446 y(for) 38 b(an) n(y) p Fm 37 w(s) p Fl 40 w(\025) p Fm 40 w(s) p Fo 999 4458 a(0) p Ft 1074 4446 a(there) g(exists) f(an) h(op) r(en) g(neigh) n(b) r(orho) r(o) r(d) f(of) h(the) g(origin) p Fl 37 w(U) p Fn 2961 4458 a(s) p Fo(+) p Fn(d) p Fe 3078 4466 a(r) p Fl 3155 4446 a(\032) i(P) p Fn 3318 4458 a(s) p Fo(+) p Fn(d) p Fe 3435 4466 a(r) p Ft 471 4554 a(suc) n(h) 31 b(that) g(the) g (Hamiltonian) f(v) n(ector) g(\014eld) h(of) p Fm 30 w(H) p Ft 38 w(is) f(de\014ned) i(on) p Fl 30 w(U) p Fn 2657 4566 a(s) p Fo(+) p Fn(d) p Fe 2774 4574 a(r) p Ft 2842 4554 a(and) f(ful\014lls) p Fm 31 w(X) p Fn 3326 4566 a(H) p Fl 3417 4554 a(2) p Fm 471 4662 a(C) p Fn 536 4632 a(r) p Fo 2 w(+2) p Ft 657 4662 a(\() p Fl(U) p Fn 741 4674 a(s) p Fo(+) p Fn(d) p Fe 858 4682 a(r) p Fm 896 4662 a(;) p Fl 14 w(P) p Fn 991 4674 a(s) p Ft 1026 4662 a(\).) p Fs 172 4838 a(De\014nition) e(3.1.) p Ft 40 w(Let) p Fm 25 w(N) p Ft 35 w(b) r(e) d(a) f(p) r(ositiv) n(e) g (in) n(teger,) g(a) g(functional) p Fm 25 w(Z) p Ft 32 w(will) g(b) r(e) h(said) f(to) g(b) r(e) h(in) p Fm 26 w(N) p Ft 9 w(-normal) 172 4946 y(form) i(if) g(it) g(is) f(indep) r (enden) n(t) i(of) e(all) h(the) g(v) -5 b(ariables) p Fl 26 w(f) p Fm(p) p Fn 1863 4958 a(j) p Fm 1897 4946 a(;) 14 b(q) p Fn 1971 4958 a(j) p Fl 2006 4946 a(g) p Fn 2048 4958 a(j) s(>N) p Ft 2221 4946 a(and) p Fl 1579 5106 a(f) p Fm -1 w(Z) q(;) g(H) p Fo 1784 5118 a(0) p Fl 1822 5106 a(g) 22 b(\021) p Ft 23 w(0) p Fm 27 w(:) p Ft 1259 w(\(10\)) 297 5283 y(F) -7 b(or) 27 b(\014xed) p Fm 28 w(N) p Ft 36 w(w) n(e) g(will) h(denote) p Fm 1331 5495 a(H) p Fo 1407 5452 a(\() p Fn(N) p Fo 6 w(\)) 1400 5517 y(0) p Ft 1522 5495 a(\() p Fm(p;) 14 b(q) p Ft 3 w(\)) 23 b(=) p Fn 1846 5391 a(N) p Fh 1816 5416 a(X) p Fn 1818 5593 a(j) p Fo 3 w(=1) p Fm 1949 5495 a(!) p Fn 2001 5507 a(j) p Fm 2046 5430 a(p) p Fo 2088 5400 a(2) p Fn 2088 5452 a(j) p Ft 2144 5430 a(+) p Fm 18 w(q) p Fo 2267 5400 a(2) p Fn 2264 5452 a(j) p 2046 5476 258 4 v Ft 2154 5552 a(2) 3325 5495 y(\(11\)) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 172 636 a(4) p Fs 172 856 a(Theorem) 41 b(3.1.) p Fg 41 w(Assume) c(H1-H2) g(and) i(\014x) e(a) h(p) l(ositive) p Fm 39 w(M) p Fg 9 w(.) 62 b(Then) 38 b(ther) l(e) g(exist) g(c) l (onstants) p Fm 37 w(s) p Fk 3282 826 a(0) p Fg 3305 856 a(,) p Fm 40 w(s) p Fk 3409 868 a(\003) p Fg 3447 856 a(,) p Fm 172 964 a(R) p Fk 235 976 a(\003) p Fm 297 964 a(>) p Ft 22 w(0) p Fg(,) 28 b(a) g(function) p Fm 27 w(N) p Ft 9 w(\() p Fm(\017) p Ft(\)) p Fg(,) g(and) g(an) g (analytic) g(c) l(anonic) l(al) h(tr) l(ansformation) p Fl 28 w(T) p Ft 44 w(:) p Fm 23 w(B) p Fn 2812 976 a(s) p Fc 2843 984 a(\003) p Ft 2882 964 a(\() p Fm(R) p Fk 2977 976 a(\003) p Ft 3016 964 a(\)) p Fl 23 w(!) 23 b(P) p Fn 3235 976 a(s) p Fc 3266 984 a(\003) p Fg 3332 964 a(that) 172 1072 y(puts) 30 b(the) g(Hamiltonian) g(in) g(normal) g (form) h(up) f(to) f(or) l(der) p Fm 31 w(M) p Fg 9 w(.) 38 b(Pr) l(e) l(cisely) 31 b(such) f(that) p Fm 1394 1233 a(H) p Fl 25 w(\016) 18 b(T) p Ft 44 w(=) p Fm 23 w(H) p Fo 1801 1190 a(\() p Fn(N) p Fo 6 w(\)) 1794 1256 y(0) p Ft 1934 1233 a(+) p Fm 18 w(Z) p Ft 24 w(+) p Fl 18 w(R) p Ft 1074 w(\(12\)) p Fg 172 1395 a(wher) l(e) p Fm 37 w(Z) p Fg 42 w(is) 36 b(in) p Fm 37 w(N) p Ft 9 w(\() p Fm(\017) p Ft(\)) p Fg(-normal) g(form.) 59 b(Mor) l(e) l(over,) 39 b(for) e(any) p Fm 37 w(s) d(>) g(s) p Fk 2379 1407 a(\003) p Fg 2453 1395 a(ther) l(e) j(exists) p Fm 35 w(R) p Fn 2963 1407 a(s) p Fg 3035 1395 a(such) f(that) p Fl 36 w(T) p Fg 172 1503 a(r) l(estricts) 28 b(to) f(an) h(analytic) h(map) f (fr) l(om) p Fm 29 w(B) p Fn 1455 1515 a(s) p Ft 1490 1503 a(\() p Fm(R) p Fn 1585 1515 a(s) p Ft 1621 1503 a(\)) p Fg 28 w(to) p Fl 27 w(P) p Fn 1836 1515 a(s) p Fg 1899 1503 a(and,) h(for) g(any) p Fm 28 w(z) p Fl 26 w(2) p Fm 23 w(B) p Fn 2578 1515 a(s) p Fo(+) p Fn(s) p Fc 2691 1499 a(0) p Ft 2719 1503 a(\() p Fm(R) p Fn 2814 1515 a(s) p Fo(+) p Fn(s) p Fc 2927 1499 a(0) p Ft 2954 1503 a(\)) p Fg 28 w(the) f(fol) t(lowing) 172 1611 y(estimate) i (holds) p Fl 1401 1772 a(k) p Fm -1 w(X) p Fk 1511 1784 a(R) p Ft 1572 1772 a(\() p Fm(z) p Ft 4 w(\)) p Fl(k) p Fn 1720 1797 a(s) p Fl 1779 1772 a(\024) p Fm 23 w(C) p Fn 1926 1784 a(s) p Fl 1961 1772 a(k) p Fm(z) p Fl 4 w(k) p Fn 2088 1737 a(M) p Fo 6 w(+1) p Fn 2088 1797 a(s) p Fo(+) p Fn(s) p Fc 2201 1780 a(0) p Ft 3325 1772 a(\(13\)) 297 1949 y(The) 24 b(pro) r(of) f(is) g(obtained) h(in) g (three) f(steps:) 35 b(First) 24 b(mak) n(e) f(a) g(Galerkin) g (cuto\013) h(of) f(the) i(original) d(system,) 172 2057 y(second) 32 b(put) h(in) f(normal) f(form) h(the) g(\014nite) h (dimensional) e(cuto\013ed) i(system,) g(third) f(c) n(ho) r(ose) f (the) h(cuto\013) 172 2165 y(so) f(that) g(the) g(error) e(due) i(to) f (the) i(Galerkin) e(appro) n(ximation) e(is) j(of) g(the) g(same) f (order) g(of) g(magnitude) h(as) 172 2273 y(the) d(size) g(of) f(the) h (reminder) f(of) h(the) g(Birkho\013) f(normal) f(form.) 297 2381 y(Concerning) g(theorem) h(3.1) f(note) i(in) f(particular) f (that) i(the) g(reminder) f(is) g(small) g(in) g(the) h(norm) f(of) p Fl 27 w(P) p Fn 3437 2393 a(s) p Ft 172 2489 a(when) k(it) f(is) g(ev) -5 b(aluated) 30 b(at) g(a) g(p) r(oin) n(t) g(of) p Fl 30 w(P) p Fn 1487 2501 a(s) p Fo(+) p Fn(s) p Fc 1600 2485 a(0) p Ft 1627 2489 a(.) 45 b(Here) p Fm 30 w(s) p Fk 1933 2459 a(0) p Ft 1986 2489 a(is) 30 b(a) g(large) p Fm 29 w(M) p Ft 38 w(dep) r(enden) n(t) h(n) n(um) n(b) r(er.) 45 b(So,) 30 b(the) 172 2597 y(situation) h(is) f(that) h(the) g(reminder) f(is) g(small) g(pro) n(vided) g(it) h(is) f(considered) g(as) g(a) g (`di\013eren) n(tial) g(op) r(erator') 172 2705 y(of) 40 b(large) e(order.) 72 b(F) -7 b(or) 40 b(this) g(reason) e(it) i(is) g (imp) r(ossible) f(to) h(deduce) g(from) f(\(12\)) h(an) n(y) f(b) r (ound) h(on) f(the) 172 2813 y(norm) 24 b(of) h(the) f(solution.) 36 b(F) -7 b(urthermore) 23 b(it) i(is) f(non) g(trivial) g(to) h(deduce) f (some) g(dynamical) g(consequences.) 172 2921 y(Nev) n(ertheless,) 40 b(adding) e(some) f(assumptions) h(one) f(can) h(compare) f(the) h (solution) g(of) g(the) h(normalized) 172 3029 y(system) 24 b(to) f(the) h(solution) f(of) g(the) h(original) e(one.) 35 b(T) -7 b(o) 23 b(this) h(end) g(w) n(e) f(will) h(restrict) f(to) g (quasilinear) f(systems) 172 3136 y(and) 28 b(assume) f(that) h(the) g (Ly) n(apuno) n(v) e(exp) r(onen) n(ts) h(are) f(of) i(order) p Fm 26 w(\017) p Ft 28 w(in) f(a) h(ball) f(of) h(radius) p Fm 26 w(\017) p Ft(.) 297 3244 y(Denote) f(b) n(y) p Fl 27 w(B) p Ft 3 w(\() p Fl(P) p Fn 844 3256 a(s) p Fo(+) p Fn(d) p Fm 964 3244 a(;) p Fl 14 w(P) p Fn 1059 3256 a(s) p Ft 1094 3244 a(\)) h(the) f(space) g(of) g(b) r(ounded) g (linear) g(op) r(erators) e(from) p Fl 27 w(P) p Fn 2802 3256 a(s) p Fo(+) p Fn(d) p Ft 2950 3244 a(to) p Fl 27 w(P) p Fn 3109 3256 a(s) p Ft 3144 3244 a(.) 37 b(First) 27 b(of) 172 3352 y(all) h(w) n(e) f(assume) g(that) h(the) g(system) f (has) g(the) h(form) 1329 3514 y(_) p Fm -37 w(z) p Ft 26 w(=) p Fm 23 w(A) p Ft(\() p Fm(z) p Ft 4 w(\)) p Fm(z) p Ft 22 w(+) p Fm 18 w(g) p Ft 3 w(\() p Fm(z) p Ft 4 w(\)) p Fl 22 w(\021) p Fm 23 w(X) p Fn 2110 3526 a(H) p Ft 2172 3514 a(\() p Fm(z) p Ft 4 w(\)) p Fm 28 w(:) p Ft 995 w(\(14\)) 271 3678 y(H2'\)) 41 b(There) 27 b(exists) p Fm 28 w(d) p Ft(,) h(and,) f(for) g(an) n(y) p Fm 27 w(s) p Fl 23 w(\025) p Fm 23 w(s) p Fo 1692 3690 a(0) p Ft 1756 3678 a(a) h(p) r(ositiv) n(e) p Fm 27 w(R) p Fn 2196 3690 a(s) p Fo(+) p Fn(d) p Ft 2317 3678 a(,) g(suc) n(h) f(that) h(the) g(map) p Fm 1255 3840 a(B) p Fn 1318 3852 a(s) p Fo(+) p Fn(d) p Ft 1439 3840 a(\() p Fm(R) p Fn 1534 3852 a(s) p Fo(+) p Fn(d) p Ft 1655 3840 a(\)) p Fl 23 w(3) p Fm 24 w(z) p Fl 26 w(7!) p Fm 23 w(A) p Ft(\() p Fm(z) p Ft 4 w(\)) p Fl 23 w(2) c(B) p Ft 3 w(\() p Fl(P) p Fn 2379 3852 a(s) p Fo(+) p Fn(d) p Fm 2499 3840 a(;) p Fl 14 w(P) p Fn 2594 3852 a(s) p Ft 2629 3840 a(\)) 471 4001 y(is) k(of) f(class) p Fm 27 w(C) p Fk 909 3971 a(1) p Ft 980 4001 a(.) 37 b(Moreo) n(v) n(er) p Fm 24 w(g) p Ft 31 w(is) 27 b(smo) r(oth,) h(i.e.) p Fm 37 w(g) p Fl 25 w(2) p Fm 24 w(C) p Fk 2227 3971 a(1) p Ft 2297 4001 a(\() p Fm(B) p Fn 2392 4013 a(s) p Fo(+) p Fn(d) p Ft 2513 4001 a(\() p Fm(R) p Fn 2608 4013 a(s) p Fo(+) p Fn(d) p Ft 2730 4001 a(\)) p Fm(;) p Fl 14 w(P) p Fn 2857 4013 a(s) p Fo(+) p Fn(d) p Ft 2978 4001 a(\).) 297 4166 y(F) -7 b(or) 49 b(an) n(y) p Fm 49 w(\017) p Ft 49 w(small) g(enough) g(consider) f(the) i(set) g(of) f(the) h (functions) p Fm 50 w(\020) p Fl 66 w(2) p Fm 60 w(C) p Fo 2882 4136 a(0) p Ft 2919 4166 a(\([0) p Fm(;) 14 b(T) p Ft 12 w(]) p Fm(;) p Fl 14 w(P) p Fn 3232 4178 a(s) p Fo(+) p Fn(d) p Ft 3352 4166 a(\)) p Fl 33 w(\\) p Fm 172 4274 a(C) p Fo 237 4244 a(1) p Ft 275 4274 a(\([0) p Fm(;) g(T) p Ft 12 w(]) p Fm(;) p Fl 14 w(P) p Fn 588 4286 a(s) p Ft 622 4274 a(\)) 28 b(ful\014lling) 1216 4452 y(sup) p Fn 1174 4526 a(t) p Fk(2) p Fo([0) p Fn(;T) p Fo 9 w(]) p Fl 1397 4452 a(k) p Fm -1 w(\020) p Ft 6 w(\() p Fm(t) p Ft(\)) p Fl(k) p Fn 1617 4477 a(s) p Fo(+) p Fn(d) p Ft 1756 4452 a(+) 60 b(sup) p Fn 1839 4526 a(t) p Fk(2) p Fo([0) p Fn(;T) p Fo 9 w(]) p Fh 2061 4356 a(\015) 2061 4406 y(\015) 2061 4456 y(\015) p Ft 2124 4430 a(_) p Fm 2107 4452 a(\020) p Ft 7 w(\() p Fm(t) p Ft(\)) p Fh 2244 4356 a(\015) 2244 4406 y(\015) 2244 4456 y(\015) p Fn 2291 4510 a(s) p Fl 2349 4452 a(\024) p Fm 23 w(\017) p Ft 854 w(\(15\)) 172 4668 y(and) 28 b(the) g(linear) f (time) h(dep) r(enden) n(t) g(equation) 1607 4829 y(_) p Fm -37 w(z) p Ft 26 w(=) p Fm 23 w(A) p Ft(\() p Fm(\020) p Ft 6 w(\() p Fm(t) p Ft(\)\)) p Fm(z) p Ft 1278 w(\(16\)) 294 4994 y(H3\)) 41 b(There) 29 b(exists) p Fm 30 w(\027) p Fl 31 w(\025) p Ft 26 w(1) h(suc) n(h) f(that) h(the) g(ev) n(olution) f (op) r(erator) p Fm 28 w(U) p Ft 9 w(\() p Fm(t;) 14 b(s) p Ft(\)) 29 b(asso) r(ciated) g(to) g(equation) 471 5102 y(\(16\)) f(exists) f(and) g(ful\014lls) h(the) g(estimate) 1348 5263 y(sup) p Fo 1261 5333 a(0) p Fk(\024) p Fn(t) p Fk(\024) p Fn(\034) p Fk 7 w(\024) p Fn(T) p Fl 1574 5263 a(k) p Fm -1 w(U) p Ft 9 w(\() p Fm(t;) 14 b(\034) p Ft 9 w(\)) p Fl(k) p Fn 1900 5288 a(`) p Fd 1928 5268 a(2) p Fe 1928 5308 a(s) p Fd(+) p Fe(d) p Fk 2033 5288 a(!) p Fn(`) p Fd 2127 5268 a(2) p Fe 2127 5308 a(s) p Fd(+) p Fe(d) p Fl 2259 5263 a(\024) p Fm 23 w(M) 9 b(e) p Fn 2476 5229 a(\014) s(\017) p Fe 2545 5204 a(\027) p Fn 2580 5229 a(T) p Fm 2660 5263 a(;) p Ft 642 w(\(17\)) 471 5472 y(with) 28 b(some) f(constan) n(ts) p Fm 27 w(M) t(;) 14 b(\014) p Ft 32 w(indep) r(enden) n(t) 28 b(of) p Fm 28 w(\020) 6 b(;) 14 b(T) 7 b(;) 14 b(\017) p Ft(.) 294 5580 y(H4\)) p Fm 41 w(g) p Ft 31 w(has) 27 b(a) g(zero) f(of) i(order) e(at) i(least) p Fm 27 w(\027) p Ft 24 w(+) 18 b(1) 27 b(at) g(the) h(origin.) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 3437 636 a(5) p Ft 297 856 a(Due) 33 b(to) f(the) h(form) f (of) g(the) h(Hamiltonian,) g(one) f(has) p Fm 32 w(A) p Ft(\() p Fm(z) p Ft 4 w(\)) f(=) p Fm 30 w(A) p Fo 2351 868 a(0) p Ft 2410 856 a(+) p Fm 21 w(B) p Ft 4 w(\() p Fm(z) p Ft 4 w(\),) j(with) p Fm 33 w(A) p Fo 2983 868 a(0) p Ft 3051 856 a(=) p Fm 31 w(X) p Fn 3216 868 a(H) p Fd 3270 876 a(0) p Ft 3339 856 a(and) p Fm 172 964 a(B) p Ft 4 w(\() p Fm(:) p Ft(\)) f(an) f(op) r(erator) e(v) -5 b(alued) 31 b(map) h(v) -5 b(anishing) 31 b(at) h(zero.) 48 b(In) 32 b(assumption) g(H3) f(w) n(e) h(are) e(thinking) i(of) g(the) 172 1072 y(case) 26 b(where) p Fm 26 w(B) p Ft 31 w(has) g(a) g(zero) f (of) h(order) p Fm 25 w(\027) p Ft 32 w(at) h(the) f(origin.) 36 b(In) 26 b(the) h(semilinear) f(case) g(one) g(has) p Fm 26 w(B) p Fl 27 w(\021) p Ft 22 w(0,) h(and) 172 1180 y(therefore) g(H3) h(is) f(automatic.) p Fs 172 1367 a(Remark) i(3.1.) p Ft 40 w(Using) d(Kato's) e(theory) p Fo 1450 1337 a(14) p Ft 1546 1367 a(one) h(can) h(pro) n(v) n(e) e (that) i(under) g(the) g(assumptions) f(H2',H3,H4) 172 1475 y(the) j(dynamics) g(of) f(the) h(system) f(is) h(lo) r(cally) f (w) n(ell) g(p) r(osed) h(in) f(an) n(y) g(space) p Fl 27 w(P) p Fn 2477 1487 a(s) p Ft 2540 1475 a(with) p Fm 28 w(s) p Fl 23 w(\025) p Fm 23 w(s) p Fo 2918 1487 a(0) p Ft 2973 1475 a(+) p Fm 18 w(d) p Ft(.) 297 1662 y(W) -7 b(e) 35 b(\014x) g(no) n(w) f(the) h(notations) f(concerning) g (the) h(ob) 5 b(jects) 34 b(w) n(e) h(are) f(going) f(to) i(compare.) 57 b(Giv) n(en) 35 b(an) 172 1770 y(initial) k(datum) p Fm 38 w(z) p Fo 735 1782 a(0) p Ft 810 1770 a(w) n(e) f(consider) e (the) j(corresp) r(onding) d(solution) p Fm 37 w(z) p Ft 4 w(\() p Fm(t) p Ft(\)) i(of) g(the) h(equations) e(of) h(motion) 172 1878 y(\(14\)) 31 b(of) g(the) h(original) e(system.) 47 b(Then) 31 b(w) n(e) g(consider) f(the) i(solution) p Fm 30 w(z) p Fn 2394 1890 a(N) p Ft 2457 1878 a(\() p Fm(t) p Ft(\)) g(of) f(the) g(\014nite) h(dimensional) 172 1986 y(normalized) 27 b(system) 1426 2161 y(_) p Fm -37 w(z) p Ft 27 w(=) p Fm 22 w(X) p Fn 1634 2194 a(H) p Fd 1692 2164 a(\() p Fe(N) p Fd 5 w(\)) 1688 2212 y(0) p Ft 1795 2161 a(\() p Fm(z) p Ft 4 w(\)) 19 b(+) p Fm 18 w(X) p Fn 2073 2173 a(Z) p Ft 2126 2161 a(\() p Fm(z) p Ft 4 w(\)) 172 2355 y(with) 26 b(initial) f(datum) g(\005) p Fn 918 2367 a(N) p Fl 981 2355 a(T) p Fk 1047 2325 a(\000) p Fo(1) p Ft 1137 2355 a(\() p Fm(z) p Fo 1208 2367 a(0) p Ft 1245 2355 a(\),) g(where) g(\005) p Fn 1625 2367 a(N) p Ft 1713 2355 a(is) f(the) i(pro) 5 b(jector) 23 b(on) h(the) h(\014rst) p Fm 25 w(N) p Ft 33 w(mo) r(des.) 36 b(Finally) 25 b(w) n(e) 172 2463 y(de\014ne) f(the) f(appro) n(ximating) f(solution) p Fm 22 w(z) p Fn 1444 2475 a(a) p Ft 1484 2463 a(\() p Fm(t) p Ft(\)) i(:=) p Fl 22 w(T) p Ft 22 w(\() p Fm(z) p Fn 1850 2475 a(N) p Ft 1913 2463 a(\() p Fm(t) p Ft(\)\),) h(whic) n(h) e(is) g(the) g(solution) g(of) g(the) h(normalized) 172 2571 y(system) k(in) g(the) g(original) e(co) r(ordinates.) p Fs 172 2759 a(Theorem) 33 b(3.2.) p Fg 41 w(Assume) d(H1,H2',H3,H4) k (and) e(\014x) p Fm 30 w(M) p Fl 35 w(\025) p Ft 25 w(2) p Fg(,) g(then) f(ther) l(e) g(exists) p Fm 31 w(s) p Fk 2878 2729 a(0) p Fm 2927 2759 a(>>) p Ft 25 w(1) p Fg(,) h(and,) h(for) 172 2867 y(any) p Fm 26 w(s) p Fg 26 w(lar) l(ge) 26 b(enough,) h(ther) l(e) f(exists) p Fm 25 w(\017) p Fn 1352 2879 a(s) p Fg 1413 2867 a(with) g(the) g(fol) t(lowing) i(pr) l (op) l(erty.) 38 b(If) 26 b(the) g(initial) h(datum) e(is) h(smo) l (oth) 172 2975 y(and) 31 b(smal) t(l) f(enough,) h(pr) l(e) l(cisely) g (if) p Fm 1499 3150 a(\017) p Ft 23 w(:=) p Fl 23 w(k) p Fm -1 w(z) p Fo 1747 3162 a(0) p Fl 1784 3150 a(k) p Fn 1826 3175 a(s) p Fo(+) p Fn(s) p Fc 1939 3158 a(0) p Fl 1989 3150 a(\024) p Fm 22 w(\017) p Fn 2110 3162 a(s) p Ft 3325 3150 a(\(18\)) p Fg 172 3325 a(then) p Fm 1128 3510 a(d) p Fn 1171 3522 a(s) p Ft 1206 3510 a(\() p Fm(z) p Ft 4 w(\() p Fm(t) p Ft(\)) p Fm(;) 14 b(z) p Fn 1451 3522 a(a) p Ft 1491 3510 a(\() p Fm(t) p Ft(\)\)) p Fl 24 w(\024) p Fm 23 w(C) 6 b(\017) p Fn 1828 3476 a(M) p Ft 1986 3510 a(for) p Fl 84 w(j) p Fm(t) p Fl(j) 24 b(\024) p Ft 2416 3454 a(1) p 2367 3491 141 4 v Fm 2367 3567 a(C) 6 b(\017) p Fn 2466 3543 a(\027) p Ft 3325 3510 a(\(19\)) p Fg 172 3705 a(wher) l(e) p Fm 31 w(d) p Fn 450 3717 a(s) p Ft 486 3705 a(\() p Fm(:;) 14 b(:) p Ft(\)) p Fg 30 w(is) 30 b(the) g(distanc) l(e) g(in) g(the) g(norm) g(of) p Fl 30 w(P) p Fn 1827 3717 a(s) p Fg 1863 3705 a(.) p Fs 172 3892 a(Remark) 35 b(3.2.) p Ft 40 w(The) c(ab) r(o) n(v) n(e) f (theorem) g(is) h(essen) n(tially) f(an) h(a) n(v) n(eraging) d (theorem,) j(indeed) h(the) f(time) h(of) 172 4000 y(v) -5 b(alidit) n(y) 29 b(of) g(its) g(description) f(of) h(the) g(dynamics) f (is) h(of) g(the) g(same) f(order) f(of) i(magnitude) g(as) f(the) h (in) n(v) n(erse) 172 4108 y(of) f(the) g(size) f(of) h(the) g(p) r (erturbation.) p Fs 172 4295 a(Remark) f(3.3.) p Ft 40 w(When) e(the) g(frequencies) f(are) f(nonresonan) n(t) f(the) j (solution) f(of) g(the) h(normalized) e(system) 172 4403 y(lie) 36 b(on) e(an) h(in) n(v) -5 b(arian) n(t) 34 b(torus,) i(and) f(so) f(the) i(same) e(is) h(true) g(for) p Fm 34 w(z) p Fn 2228 4415 a(a) p Ft 2268 4403 a(\() p Fm(t) p Ft(\).) 60 b(F) -7 b(rom) 34 b(this) i(one) e(can) h(conclude) 172 4511 y(that) e(there) f(exists) g(a) g(\014nite) h(dimensional) e (smo) r(oth) h(torus) p Ff 32 w(T) p Fn 2132 4523 a(z) p Fd 2164 4531 a(0) p Ft 2232 4511 a(with) h(the) f(prop) r(ert) n(y) g (that,) h(up) g(to) f(the) 172 4619 y(considered) 27 b(times) h(ones) f(has) p Fm 1435 4794 a(d) p Fn 1478 4806 a(s) p Ft 1514 4794 a(\() p Ff(T) p Fn 1602 4806 a(z) p Fd 1634 4814 a(0) p Fm 1670 4794 a(;) 14 b(z) p Ft 4 w(\() p Fm(t) p Ft(\)\)) p Fl 23 w(\024) p Fm 22 w(C) 6 b(\017) p Fn 2085 4760 a(M) p Fm 2187 4794 a(:) p Fs 172 5065 a(4.) 47 b(Applications) 172 5214 y(4.1.) p Fb 48 w(The) 35 b(water) f(wave) g(pr) -5 b(oblem) p Ft 172 5364 a(The) 38 b(w) n(ater) f(w) n(a) n(v) n(e) f(problem) i (consists) f(in) h(describing) f(the) h(motion) f(of) h(the) g(free) g (surface) f(of) h(a) f(\015uid) 172 5472 y(sub) 5 b(jected) 28 b(to) f(the) g(gra) n(vitational) d(force.) 36 b(Here) 27 b(w) n(e) g(will) g(consider) f(the) h(case) g(of) g(a) f(\015uid) i (lying) e(in) i(a) e(t) n(w) n(o) 172 5580 y(dimensional) 31 b(domain) f(of) h(in\014nite) g(depth.) 47 b(W) -7 b(e) 32 b(will) f(study) g(space) f(p) r(erio) r(dic) g(solutions.) 46 b(In) 31 b(terms) g(of) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 172 636 a(6) p Ft 172 856 a(the) g(v) n(elo) r(cit) n(y) f (p) r(oten) n(tial) p Fm 27 w(') p Ft(\() p Fm(x;) 14 b(y) p Ft 3 w(\)) 27 b(and) g(of) f(the) h(pro\014le) f(of) h(the) g (surface) p Fm 25 w(\021) p Ft 3 w(\() p Fm(x) p Ft(\),) h(the) f (equations) f(of) h(motion) 172 964 y(are) g(giv) n(en) g(b) n(y) 924 1124 y(\001) p Fm(') p Ft 35 w(=) 34 b(0) p Fm 14 w(;) p Ft 124 w(0) p Fl 22 w(\024) p Fm 23 w(x) p Fl 23 w(\024) p Ft 23 w(2) p Fm(\031) 17 b(;) p Fl 41 w(\0001) p Fm 23 w(<) 23 b(y) i(<) e(\021) p Ft 3 w(\() p Fm(x) p Ft(\)) 893 b(\(20\)) p Fm 994 1257 a(') p Fl 25 w(!) p Ft 25 w(0) p Fm 14 w(;) 373 b(y) p Fl 26 w(!) 23 b(\0001) p Ft 1371 w(\(21\)) p Fm 977 1390 a(\021) p Fn 1018 1402 a(t) p Ft 1082 1390 a(=) p Fm 34 w(') p Fn 1235 1402 a(y) p Fl 1293 1390 a(\000) p Fm 18 w(\021) p Fn 1417 1402 a(x) p Fm 1459 1390 a(') p Fn 1513 1402 a(x) p Fm 1569 1390 a(;) 97 b(') p Fn 1743 1402 a(t) p Ft 1796 1390 a(=) p Fl 23 w(\000) p Fm(g) s(\021) p Fl 20 w(\000) p Ft 18 w(\() p Fl(r) p Fm(') p Ft(\)) p Fo 2323 1355 a(2) p Fm 2376 1390 a(;) 124 b(y) p Ft 26 w(=) p Fm 23 w(\021) p Ft 3 w(\() p Fm(x) p Ft(\)) 492 b(\(22\)) p Fm 811 1522 a(\021) p Ft 3 w(\() p Fm(x;) 14 b(y) p Ft 3 w(\)) 35 b(=) p Fm 34 w(\021) p Ft 3 w(\() p Fm(x) p Ft 19 w(+) 18 b(2) p Fm(\031) s(;) c(y) p Ft 3 w(\)) p Fm 28 w(;) 96 b(') p Ft(\() p Fm(x;) 14 b(y) p Ft 3 w(\)) 24 b(=) p Fm 23 w(') p Ft(\() p Fm(x) p Ft 20 w(+) 18 b(2) p Fm(\031) s(;) c(y) p Ft 3 w(\)) 768 b(\(23\)) 172 1682 y(Zakharo) n(v) p Fo 514 1652 a(22) p Ft 608 1682 a(p) r(oin) n(ted) 26 b(out) g(that) g (this) g(is) g(a) f(Hamiltonian) h(system.) 36 b(The) 26 b(corresp) r(onding) e(Hamiltonian) 172 1790 y(function) 33 b(is) e(the) h(energy) f(of) h(the) g(\015uid,) h(and) f(conjugated) f (v) -5 b(ariables) 30 b(are) h(giv) n(en) g(b) n(y) g(the) h(w) n(a) n (v) n(e) e(pro\014le) p Fm 172 1898 a(\021) p Ft 3 w(\() p Fm(x) p Ft(\)) f(and) f(the) g(v) n(elo) r(cit) n(y) e(p) r(oten) n (tial) p Fm 28 w(') p Ft(\() p Fm(x;) 14 b(\021) p Ft 3 w(\() p Fm(x) p Ft(\)\)) 30 b(at) e(the) g(free) f(surface.) 297 2006 y(It) 34 b(w) n(as) f(sho) n(wn) g(b) n(y) g(Zakharo) n(v) e(that) j(close) f(to) h(the) g(equilibrium) f(solution) h(in) g(whic) n(h) f (the) h(\015uid) h(is) 172 2114 y(at) f(rest) f(and) h(the) g(surface) e (is) i(horizon) n(tal) e(the) i(Hamiltonian) f(has) g(the) h(form) g (\(1\)) f(with) p Fm 35 w(!) p Fn 3061 2126 a(k) p Ft 3134 2114 a(=) p Fh 3232 2043 a(p) p 3315 2043 135 4 v Fm 3315 2114 a(g) p Fl 3 w(j) p Fm(k) p Fl 3 w(j) p Ft -1 w(,) p Fm 172 2222 a(k) p Fl 44 w(2) p Ff 41 w(Z) p Fl -1 w(nf) p Ft -1 w(0) p Fl(g) p Ft -1 w(.) 62 b(So) 38 b(it) h(is) f(easy) f(to) h(v) n(erify) f(that) h(the) h(frequencies) e (ful\014ll) i(the) g(assumption) e(H1.) 68 b(The) 172 2330 y(regularit) n(y) 32 b(of) h(the) h(nonlinear) f(part) p Fm 33 w(P) p Ft 45 w(can) g(b) r(e) h(pro) n(v) n(ed) e(b) n(y) h (classical) f(theory) h(of) g(elliptic) h(equations.) 172 2438 y(So) c(assumption) g(H2) g(holds.) 45 b(It) 30 b(follo) n(ws) g(that) g(according) f(to) h(theorem) g(2.1) f(there) h (exists) g(a) g(canonical) 172 2546 y(transformation) c(putting) j(the) f(Hamiltonian) f(in) h(normal) e(form) i(up) g(to) f(an) n(y) g (\014nite) h(order.) 297 2654 y(W) -7 b(e) 29 b(p) r(oin) n(t) g(out) f (that) h(in) f(this) h(problem) f(nothing) h(is) f(kno) n(wn) g(on) g (the) h(Ly) n(apuno) n(v) e(exp) r(onen) n(ts) h(of) g(the) 172 2762 y(system,) f(and) g(therefore) f(w) n(e) g(are) g(not) h(able) f (to) h(deduce) g(an) n(y) f(\(rigorous\)) f(conclusion) h(on) h(the) g (dynamics) 172 2869 y(of) h(the) g(system.) 297 2977 y(The) g(\014rst) g(terms) g(of) g(the) h(function) p Fm 29 w(Z) p Ft 33 w(w) n(ere) f(computed) g(explicitly) g(in) h(a) f (series) f(of) h(pap) r(er) p Fo 3155 2947 a(13) p Fn(;) p Fo(9) p Fn(;) p Fo(10) p Fn(;) p Fo(12) p Ft 3449 2977 a(,) 172 3085 y(with) 38 b(the) f(surprising) e(outcome) i(that) g(up) g (to) g(order) e(four) p Fm 37 w(H) p Fo 2165 3097 a(0) p Ft 2226 3085 a(+) p Fm 25 w(Z) p Ft 42 w(is) i(in) n(tegrable.) 63 b(The) 37 b(dynamics) 172 3193 y(of) d(this) f(in) n(tegrable) f (system) h(has) g(also) f(b) r(een) i(studied) f(in) h(detail.) 54 b(Ho) n(w) n(ev) n(er) 31 b(as) i(far) f(as) h(w) n(e) g(kno) n(w) f (the) 172 3301 y(rigorous) f(existence) h(of) h(the) g(normalizing) f (transformation) f(w) n(as,) i(up) g(to) g(no) n(w,) g(established) g (only) f(for) 172 3409 y(the) c(transformation) e(putting) j(the) f (system) f(in) h(third) g(order) e(normal) g(form) p Fo 2567 3379 a(9) p Ft 2604 3409 a(.) p Fs 172 3668 a(4.2.) p Fb 48 w(A) 34 b(quasiline) -5 b(ar) 34 b(wave) g(e) -5 b(quation) p Ft 172 3818 a(Consider) 27 b(the) p Fm 28 w(n) p Ft 28 w(dimensional) g(parallelepip) r(ed) p Fl 27 w(R) p Ft 28 w(with) h(sides) f(of) h(length) p Fm 27 w(L) p Fn 2621 3830 a(i) p Ft 2648 3818 a(,) g(namely) p Fl 1031 3977 a(R) p Ft 23 w(:=) p Fl 23 w(f) p Fm -1 w(x) p Fl 24 w(\021) p Ft 22 w(\() p Fm(x) p Fo 1513 3989 a(1) p Fm 1551 3977 a(;) 14 b(:::;) g(x) p Fn 1741 3989 a(n) p Ft 1787 3977 a(\)) p Fl 23 w(2) p Ff 24 w(R) p Fn 1975 3943 a(n) p Ft 2077 3977 a(:) 51 b(0) p Fm 22 w(<) 23 b(x) p Fn 2350 3989 a(i) p Fm 2401 3977 a(<) g(L) p Fn 2546 3989 a(i) p Fl 2573 3977 a(g) p Ft 172 4137 a(and) 28 b(the) g(formal) f(Hamiltonian) g(system) g(de\014ned) h(b) n(y) g(the) g(Hamiltonian) p Fm 826 4359 a(H) p Ft 7 w(\() p Fm(u;) 14 b(v) p Ft 3 w(\)) 23 b(=) p Fh 1205 4246 a(Z) p Fk 1251 4435 a(R) p Fh 1326 4217 a(") p Fm 1385 4303 a(v) p Fo 1428 4273 a(2) p 1385 4340 81 4 v Ft 1404 4416 a(2) 1493 4359 y(+) p Fl 1586 4303 a(kr) p Fm(u) p Fl(k) p Fo 1786 4261 a(2) p 1586 4340 237 4 v Ft 1684 4416 a(2) 1852 4359 y(+) p Fm 1945 4303 a(mu) p Fo 2066 4273 a(2) p 1945 4340 158 4 v Ft 2003 4416 a(2) 2131 4359 y(+) p Fm 18 w(W) p Ft 12 w(\() p Fm(u;) p Fl 14 w(r) p Fm(u) p Ft(\)) p Fh 2570 4217 a(#) p Fm 2632 4359 a(d) p Fn 2675 4325 a(n) p Fm 2720 4359 a(x) 28 b(;) p Ft 507 w(\(24\)) 172 4581 y(where) p Fm 29 w(u) p Ft 29 w(is) i(a) f(function) h(on) p Fl 29 w(R) p Ft 29 w(v) -5 b(anishing) 29 b(at) h(its) f(b) r(oundary) -7 b(,) p Fm 29 w(v) p Ft 30 w(=) 40 b(_) p Fm -38 w(u) p Ft 29 w(is) 30 b(the) f(momen) n(tum) h(conjugated) 172 4689 y(to) p Fm 28 w(u) p Ft(,) d(and) p Fm 28 w(W) p Ft 39 w(is) h(a) f(smo) r(oth) g(function.) 38 b(The) 27 b(corresp) r(onding) f(Hamilton) i(equations) e(ha) n(v) n(e) h(the) h (form) p Fm 941 4849 a(u) p Fn 989 4861 a(tt) p Fl 1061 4849 a(\000) p Ft 18 w(\001) p Fm(u) p Ft 18 w(+) p Fm 18 w(mu) p Fl 18 w(\000) p Fm 18 w(b) p Fn 1620 4861 a(ij) p Ft 1678 4849 a(\() p Fm(u;) p Fl 14 w(r) p Fm(u) p Ft(\)) p Fm(@) p Fn 1988 4861 a(i) p Fm 2016 4849 a(@) p Fn 2060 4861 a(j) p Fm 2095 4849 a(u) p Ft 18 w(+) p Fm 18 w(g) p Ft 3 w(\() p Fm(u;) p Fl 14 w(r) p Fm(u) p Ft(\)) 22 b(=) h(0) 620 b(\(25\)) 172 5009 y(\(with) 34 b(summation) e(con) n(v) n(en) n(tion) f(o) n(v) n(er) g(the) i (indexes) p Fm 32 w(i;) 14 b(j) p Ft 37 w(and) 32 b(suitable) g (functions) p Fm 33 w(b) p Fn 2898 5021 a(ij) p Ft 2989 5009 a(and) p Fm 32 w(g) p Ft 3 w(\)) h(whic) n(h) 172 5117 y(should) i(b) r(e) g(supplemen) n(ted) g(with) g(Diric) n(hlet) f (b) r(oundary) g(conditions.) 57 b(In) 35 b(order) e(to) i(\014t) g (the) g(abstract) 172 5225 y(sc) n(heme) f(w) n(e) f(in) n(tro) r(duce) g(the) h(phase) f(spaces) p Fl 33 w(F) p Fn 1673 5237 a(s) p Ft 1742 5225 a(comp) r(osed) g(of) g(the) i(functions) e(\() p Fm(u;) 14 b(v) p Ft 3 w(\)) p Fl 34 w(2) p Fm 33 w(H) p Fo 3134 5194 a(~) p Fn -35 w(s) p Fo(+1) p Fl 3274 5225 a(\010) p Fm 22 w(H) p Fo 3439 5194 a(~) p Fn -35 w(s) p Ft 172 5332 a(with) 32 b(~) p Fm -45 w(s) p Ft 23 w(:=) p Fm 22 w(ns=) p Ft(2,) 27 b(ful\014lling) h(the) g(compatibilit) n(y) f (conditions) 361 5534 y(\() p Fl(\000) p Ft(\001\)) p Fn 559 5500 a(j) p Fm 594 5534 a(u) p Fh 642 5463 a(\014) 642 5513 y(\014) p Fn 669 5567 a(@) p Fo 4 w(\012) p Ft 783 5534 a(=) 22 b(0) p Fm 28 w(;) p Ft 96 w(0) p Fl 23 w(\024) p Fm 23 w(j) p Fl 28 w(\024) p Fh 1361 5417 a(\024) p Ft 1420 5478 a(~) p Fm -46 w(s) p 1415 5515 42 4 v Ft 1415 5591 a(2) p Fh 1467 5417 a(\025) p Ft 1552 5534 a(;) 180 b(\() p Fl(\000) p Ft(\001\)) p Fn 1953 5500 a(j) p Fm 1988 5534 a(v) p Fh 2031 5463 a(\014) 2031 5513 y(\014) p Fn 2059 5567 a(@) p Fo 4 w(\012) p Ft 2173 5534 a(=) 22 b(0) p Fm 28 w(;) p Ft 96 w(0) p Fl 23 w(\024) p Fm 23 w(j) p Fl 28 w(\024) p Fh 2751 5417 a(\024) p Ft 2808 5478 a(~) p Fm -45 w(s) p Ft 18 w(+) c(1) p 2805 5515 182 4 v 2875 5591 a(2) p Fh 2997 5417 a(\025) p Fl 3059 5534 a(\000) p Ft 18 w(1) p Fm 27 w(:) p Ft 91 w(\(26\)) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 3437 636 a(7) p Ft 172 852 a(P) n(assing) f(to) h(F) -7 b(ourier) 27 b(co) r(e\016cien) n(t) g(it) i(is) e(easy) g(to) g(see) g (that) p Fl 28 w(F) p Fn 2073 864 a(s) p Ft 2136 852 a(is) h(isomorphic) e(to) p Fl 27 w(P) p Fn 2799 864 a(s) p Ft 2835 852 a(.) 297 960 y(In) 32 b(order) e(to) i(satisfy) f (the) h(nonresonance) e(condition) i(H1) f(remark) g(that) h(the) g (frequencies) f(dep) r(end) 172 1068 y(parametrically) 26 b(on) h(the) h(sides) p Fm 28 w(L) p Fn 1239 1080 a(i) p Ft 1293 1068 a(of) g(the) g(domain) f(and) h(on) f(the) h(mass) p Fm 27 w(m) 23 b(>) p Ft 22 w(0:) p Fm 992 1335 a(!) p Fo 1044 1350 a(\() p Fn(j) p Fd 1097 1358 a(1) p Fn 1130 1350 a(;::;j) p Fe 1237 1358 a(n) p Fo 1277 1350 a(\)) p Ft 1330 1335 a(=) p Fh 1418 1177 a(s) p 1501 1177 1152 4 v 1501 1218 a(\022) p Ft 1573 1279 a(2) p Fm(\031) p 1572 1316 94 4 v 1572 1392 a(L) p Fo 1629 1404 a(1) p Fm 1676 1335 a(j) p Fo 1710 1347 a(1) p Fh 1747 1218 a(\023) p Fo 1808 1235 a(2) p Ft 1864 1335 a(+) p Fm 18 w(:::) p Ft 19 w(+) p Fh 2118 1218 a(\022) p Ft 2194 1279 a(2) p Fm(\031) p 2189 1316 102 4 v 2189 1392 a(L) p Fn 2246 1404 a(n) p Fm 2300 1335 a(j) p Fn 2334 1347 a(n) p Fh 2380 1218 a(\023) p Fo 2441 1235 a(2) p Ft 2496 1335 a(+) p Fm 18 w(m) p Ft 673 w(\(27\)) p Fs 172 1592 a(Theorem) k(4.1.) p Fg 41 w([Se) l(e) p Fo 909 1562 a(2) p Fg 947 1592 a(]) g(F) -6 b(or) 27 b(any) p Fm 27 w(M) p Fg 36 w(ther) l(e) g(exists) f(a) i(set) p Fl 26 w(N) 36 b(\032) p Ff 22 w(R) p Fn 2293 1556 a(n) p Fo(+1) 2293 1613 y(+) p Fg 2455 1592 a(with) 28 b(ful) t(l) f(me) l (asur) l(e,) h(such) f(that,) 172 1700 y(if) p Ft 31 w(\() p Fm(L) p Fo 342 1712 a(1) p Fm 379 1700 a(;) 14 b(:::;) g(L) p Fn 579 1712 a(n) p Fm 624 1700 a(;) g(m) p Ft(\)) p Fl 23 w(2) 23 b(N) p Fg 42 w(then) 30 b(the) g(fr) l(e) l (quencies) g(ar) l(e) g(nonr) l(esonant) f(and) i(ful\014l) t(l) f(the) g(c) l(ondition) h(H1.) 297 1896 y(Assume) g(that) h(the) g(p) l (otential) p Fm 33 w(W) p Fg 43 w(is) h(even) f(in) g(e) l(ach) h(of) f (its) g(ar) l(guments) f(and) i(that) f(it) f(has) i(a) f(zer) l(o) h (of) 172 2003 y(or) l(der) p Fm 30 w(\027) p Ft 22 w(+) 16 b(2) p Fg 28 w(at) 29 b(the) g(origin) p Ft(,) f(then) f(H2',H3,H4) f (hold) g(and) h(therefore) f(theorem) g(3.2) f(applies.) 36 b(Consider) 172 2111 y(an) 28 b(initial) g(datum) f(of) h(the) g(form) p Fm 1148 2298 a(u) p Ft(\() p Fm(x;) p Ft 14 w(0\)) 23 b(=) p Fm 22 w(\017u) p Fo 1578 2310 a(0) p Ft 1615 2298 a(\() p Fm(x) p Ft(\)) p Fm 29 w(;) p Ft 111 w(_) p Fm -37 w(u) p Ft -1 w(\() p Fm(x;) p Ft 14 w(0\)) h(=) p Fm 22 w(\017v) p Fo 2297 2310 a(0) p Ft 2334 2298 a(\() p Fm(x) p Ft(\)) p Fm 29 w(:) p Ft 828 w(\(28\)) p Fs 172 2493 a(Theorem) f(4.2.) p Fg 41 w([Se) l(e) p Fo 905 2463 a(2) p Fg 943 2493 a(]) h(Fix) p Fm 23 w(M) p Fg 9 w(,) h(assume) e(that) h(the) g(p) l(ar) l(ameters) g(b) l(elong) g (to) p Fl 24 w(N) p Fg 35 w(and) g(that) g(the) g(p) l(otential) p Fm 172 2601 a(W) p Fg 42 w(is) 30 b(even) g(and) g(smo) l(oth;) h(then) f(ther) l(e) f(exists) p Fm 30 w(s) p Fk 1699 2571 a(0) p Fg 1752 2601 a(with) h(the) g(fol) t(lowing) i(pr) l(op) l(erty.) 40 b(If) p Fl 1359 2788 a(k) p Fm(u) p Fo 1449 2800 a(0) p Fl 1486 2788 a(k) p Fn 1527 2812 a(s) p Fo(+) p Fn(s) p Fc 1640 2796 a(0) p Fm 1711 2788 a(;) p Fl 99 w(k) p Fm -1 w(v) p Fo 1914 2800 a(0) p Fl 1952 2788 a(k) p Fn 1993 2812 a(s) p Fo(+) p Fn(s) p Fc 2106 2796 a(0) p Fl 2156 2788 a(\024) p Ft 23 w(1) p Fg 172 2974 a(then,) 30 b(for) h(smal) t(l) g(enough) p Fm 30 w(\017) p Fg 29 w(ther) l(e) f(exists) g(a) g(\014nite) f(dimensional) i(torus) p Ff 30 w(T) p Fn 2534 2986 a(z) p Fd 2566 2994 a(0) p Fn 2597 2986 a(;\017) p Fg 2678 2974 a(such) f(that) g(one) g(has) p Fm 953 3160 a(d) p Fn 996 3172 a(s) p Ft 1031 3160 a(\() p Fm(z) p Ft 4 w(\() p Fm(t) p Ft(\)) p Fm(;) p Ff 14 w(T) p Fn 1293 3172 a(z) p Fd 1325 3180 a(0) p Fn 1357 3172 a(;\017) p Ft 1408 3160 a(\)) p Fl 24 w(\024) p Fm 22 w(C) p Fn 1610 3172 a(M) p Fm 1684 3160 a(\017) p Fn 1718 3126 a(M) s(=) p Fo(2) p Fg 1940 3160 a(for) p Fl 86 w(j) p Fm(t) p Fl(j) 23 b(\024) p Ft 23 w(\() p Fm(C) p Fn 2406 3172 a(M) p Fm 2480 3160 a(\017) p Ft(\)) p Fk 2546 3126 a(\000) p Fn(\027) p Fm 2669 3160 a(:) p Ft 633 w(\(29\)) p Fs 172 3439 a(5.) 47 b(Concluding) 31 b(remarks) p Ft 172 3588 a(The) 26 b(main) g(limitation) g(of) g (theorem) g(3.2) f(rests) g(in) h(the) h(time) f(of) g(v) -5 b(alidit) n(y) 26 b(of) g(its) g(prediction.) 36 b(Indeed,) 26 b(in) 172 3696 y(the) 32 b(\014nite) f(dimensional) g(case) f(and) h (also) e(in) j(some) e(in\014nite) i(dimensional) e(cases) p Fo 2747 3666 a(3) p Fn(;) p Fo(4) p Ft 2867 3696 a(one) h(has) f(that) h (the) 172 3804 y(solution) i(remains) f(v) n(ery) g(close) g(to) g(a) h (`quasi{in) n(v) -5 b(arian) n(t') 30 b(torus) j(up) g(to) g(times) p Fl 33 w(O) p Ft 2 w(\() p Fm(\017) p Fk 2820 3774 a(\000) p Fn(M) p Fd 2935 3782 a(1) p Ft 2972 3804 a(\)) g(for) f(an) n(y) p Fm 33 w(M) p Fo 3413 3816 a(1) p Ft 3449 3804 a(.) 172 3912 y(So) 26 b(it) g(is) g(natural) f(to) h(ask) f(whether) h(a) g (similar) f(result) g(holds) h(also) f(in) h(the) g(case) f(considered) g(b) n(y) h(theorem) 172 4020 y(3.2.) 297 4128 y(T) -7 b(o) 27 b(discuss) g(this) h(p) r(oin) n(t) g(consider) f(the) h (particular) e(case) h(of) g(equation) g(\(25\)) g(giv) n(en) g(b) n(y) p Fm 1322 4314 a(u) p Fn 1370 4326 a(tt) p Fl 1442 4314 a(\000) p Ft 18 w(\(1) 18 b(+) p Fm 18 w(u) p Fn 1748 4280 a(\027) 1748 4335 y(x) p Ft 1790 4314 a(\)) p Fm(u) p Fn 1870 4326 a(xx) p Ft 1967 4314 a(+) p Fm 18 w(mu) p Ft 23 w(=) 23 b(0) 1001 b(\(30\)) 172 4500 y(with) 26 b(Diric) n(hlet) e(b) r(oundary) g(conditions) h(on) f([0) p Fm(;) 14 b(\031) p Ft 3 w(].) 36 b(According) 24 b(to) g(the) h (previous) f(theory) g(one) g(has) g(that) 172 4608 y(for) i(almost) g (all) f(v) -5 b(alues) 26 b(of) g(the) h(mass) p Fm 25 w(m) p Ft 26 w(the) g(frequencies) e(ful\014ll) i(H1.) 36 b(Theorem) 26 b(4.2) f(applies) h(pro) n(vided) p Fm 172 4716 a(\027) p Ft 33 w(is) i(ev) n(en.) 297 4824 y(No) n(w,) 23 b(it) g(has) f(b) r(een) h(pro) n(v) n(ed) e(b) n(y) i (Klainerman) e(and) h(Ma) 5 b(jda) p Fo 2132 4794 a(16) p Ft 2225 4824 a(that) 23 b(in) f(the) h(case) p Fm 22 w(m) p Ft 23 w(=) g(0) f(the) h(solution) 172 4932 y(of) 35 b(\(30\)) f(with) h(initial) f(data) g(\(28\)) g(dev) n(elops) g(a) g (singularit) n(y) f(in) h(the) h(second) f(deriv) -5 b(ativ) n(e) 33 b(at) i(a) f(time) h(of) 172 5040 y(order) p Fm 34 w(\017) p Fk 431 5010 a(\000) p Fn(\027) p Ft 524 5040 a(.) 58 b(In) 35 b(the) g(case) p Fm 33 w(m) p Ft 35 w(=) g(0) f(the) h(nonresonance) e(condition) h(of) h(our) f (theorem) g(is) g(violated,) i(so) 172 5148 y(Klainerman) 28 b(and) g(Ma) 5 b(jda's) 28 b(result) g(is) g(not) h(a) f(coun) n (terexample) f(to) i(theorem) f(3.2,) g(but) h(w) n(e) f(think) h(that) 172 5256 y(it) f(p) r(oses) f(serious) g(doubts) g(on) h(the) g(v) -5 b(alidit) n(y) 27 b(of) h(our) f(description) g(for) g(longer) f(times) i(scales.) 297 5364 y(Finally) 35 b(w) n(e) g(recall) f(that) i(up) g (to) f(no) n(w) f(v) n(ery) g(little) i(w) n(as) f(kno) n(wn) f (concerning) g(equations) h(in) g(more) 172 5472 y(than) g(one) f (space) f(dimensions) p Fo 1158 5441 a(6) p Fn(;) p Fo(8) p Fn(;) p Fo(19) p Fn(;) p Fo(4) p Ft 1421 5472 a(and) h(on) g(the) g (dynamics) g(of) g(quasilinear) f(equations.) 56 b(W) -7 b(e) 35 b(also) 172 5580 y(men) n(tion) 28 b(the) g(pap) r(ers) p Fo 875 5549 a(7) p Fn(;) p Fo(21) p Ft 1025 5580 a(that) g(are) f (strongly) f(related) h(to) g(the) h(presen) n(t) f(one.) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Ft 1286 -14 a(D.) 28 b(Bam) n(busi) 82 b(Birkho\013) 27 b(Normal) g(F) -7 b(orm) 27 b(for) g(PDEs) p -300 -14 3 200 v -557 46 200 3 v 3957 46 V 3898 -14 3 200 v -300 6305 V -557 6046 200 3 v 3898 6305 3 200 v 3960 6046 200 3 v Fp 172 636 a(8) p Fs 172 852 a(Ac) m(kno) m(wledgmen) m(ts) p Ft 172 1001 a(This) 21 b(w) n(ork) f(has) g(b) r(een) i(partially) e (supp) r(orted) h(b) n(y) f(INT) -7 b(AS-00.221) 19 b(pro) 5 b(ject) 21 b(and) f(b) n(y) h(\\Grupp) r(o) g(Nazionale) 172 1109 y(di) 28 b(Fisica) f(Matematica") g(of) g(\\Istituto) h(di) g (Alta) g(Matematica".) p Fs 172 1367 a(References) p Fr 172 1499 a(1.) 51 b(D.) 26 b(Bam) n(busi,) p Fq 25 w(Math.) i(Z.) p Fa 25 w(130) p Fr(,) e(345) h(\(1999\).) 172 1591 y(2.) 51 b(D.) c(Bam) n(busi,) k(An) 46 b(Av) n(eraging) h (Theorem) g(for) g(Quasilinear) h(Hamiltonian) f(PDEs.,) p Fq 53 w(A) n(nnales) g(Henri) 282 1682 y(Poinc) l(ar) n(\023) -37 b(e) p Fr(,) 27 b(to) f(app) r(ear.) 172 1773 y(3.) 51 b(D.) 26 b(Bam) n(busi,) p Fq 25 w(Commun.) h(Math.) h(Phys.) p Fa 26 w(234) p Fr(,) e(253) h(\(2003\).) 172 1865 y(4.) 51 b(D.) 25 b(Bam) n(busi) h(and) f(B.) h(Greb) r(ert,) g(F) -6 b(orme) 25 b(normale) h(p) r(our) f(NLS) g(en) g(dimension) g (quelconque,) p Fq 25 w(CRAS) j(Paris) p Fr(,) 282 1956 y(to) e(app) r(ear.) 172 2047 y(5.) 51 b(D.) 26 b(Bam) n(busi) f(and) g (N.) h(N.) f(Nekhoroshev,) p Fq 25 w(A) l(cta) k(Applic) l(andae) f (Mathematic) l(ae) p Fa 28 w(70) p Fr(,) e(1,) g(\(2002\).) 172 2139 y(6.) 51 b(J.) 26 b(Bourgain,) p Fq 28 w(A) n(nn.) h(Math.) p Fa 26 w(148) p Fr(,) f(363,) h(\(1998\).) 172 2230 y(7.) 51 b(J.) 26 b(Bourgain,) p Fq 28 w(J.) h(A) n(nal.) g(Math.) p Fa 26 w(80) p Fr(,) f(1,) g(\(2000\).) 172 2321 y(8.) 51 b(J.) 32 b(Bourgain,) p Fq 34 w(Gr) l(e) l(en) h(F) -6 b(unctions) 34 b(Estimates) g(for) e(L) l(attic) l(e) i(Schr\177) -39 b(odinger) 34 b(Op) l(er) l(ators) h(and) e(Applic) l(ations) p Fr 33 w(,) 282 2413 y(Preprin) n(t) 25 b(\(2003\).) 172 2504 y(9.) 51 b(W.) 23 b(Craig,) p Fq 25 w(Birkho\013) i(normal) g (form) g(for) g(water) h(waves) p Fr(,) e(Mathematical) g(problems) e (in) h(the) f(theory) g(of) i(w) n(ater) 282 2595 y(w) n(a) n(v) n(es) h (\(F.Dias,) h(J.M.Ghidaglia) i(and) c(J.C.Saut) i(ed.\),) f(Con) n (temp) r(orary) f(Mathematics,) p Fa 26 w(200) p Fr(,) h(AMS) g(1996.) 172 2687 y(10.) 51 b(W.) 26 b(Craig) h(and) f(C.) g(Sulem,) p Fq 25 w(J.) h(Comp.) g(Phys.) p Fa 26 w(108) p Fr(,) g(73,) f (\(1993\).) 172 2778 y(11.) 51 b(W.) 26 b(Craig) h(and) f(C.) g(E.) g (W) -6 b(a) n(yne,) p Fq 25 w(Comm.) 27 b(Pur) l(e) h(Appl.) f(Math.) p Fa 54 w(46) p Fr(,) f(1409,) h(\(1993\).) 172 2869 y(12.) 51 b(W.) 26 b(Craig) h(and) f(P) -6 b(.) 26 b(A.) f(W) -6 b(orfolk,) p Fq 27 w(Physic) l(a) 28 b(D) p Fa 26 w(84) p Fr(,) e(513,) h(\(1995\).) 172 2961 y(13.) 51 b(A.) 26 b(I.) g(Dy) n(ac) n(henk) n(o) e(and) h(V.) h(E.) g(Zakharo) n(v,) p Fq 26 w(Phy.) h(L) l(ett.) h(A) p Fa 26 w(190) p Fr(,) f(144,) g (\(1994\).) 172 3052 y(14.) 51 b(T.) 27 b(Kato,) p Fq 26 w(L) l(e) l(ct.) h(Notes) h(Math.) p Fa 26 w(448) p Fr(,) e(\(1975\).) 172 3143 y(15.) 51 b(T.) 30 b(Kato,) p Fq 31 w(A) n(bstr) l(act) j(Di\013er) l(ential) f(Equations) g(and) f (nonline) l(ar) g(Mixe) l(d) g(Pr) l(oblems) p Fr(.) g(Scuola) e (Normale) h(Su-) 282 3235 y(p) r(eriore,) d(Pisa,) g(1985.) 172 3326 y(16.) 51 b(S.) 26 b(Klainerman) f(and) h(A.) f(Ma) t(jda,) p Fq 28 w(Comm.) h(Pur) l(e) i(and) g(Appl.) f(Math.) p Fa 26 w(33) p Fr(,) g(241,) g(\(1980\).) 172 3417 y(17.) 51 b(T.) 27 b(Kapp) r(eler) f(and) f(J.) h(P\177) -38 b(osc) n(hel,) 27 b(KAM) f(&) f(KdV,) g(Springer,) h(2003.) 172 3509 y(18.) 51 b(S.) 26 b(B.) g(Kuksin,) p Fq 25 w(F) -6 b(unct.) 29 b(A) n(nal.) e(Appl.) p Fa 25 w(21) p Fr(,) f(192,) h(\(1987\).) 172 3600 y(19.) 51 b(S.) 26 b(B.) g(Kuksin,) p Fq 25 w(PMM) h(U.S.S.R.) p Fa 24 w(53) p Fr(,) g(150,) g(\(1989\).) 172 3691 y(20.) 51 b(S.) 26 b(B.) g(Kuksin,) p Fq 25 w(A) n(nalysis) i(of) g(Hamiltonian) f (PDEs) p Fr(.) f(Oxford) f(Univ) n(ersit) n(y) g(Press.) i(2000.) 172 3783 y(21.) 51 b(K.) 26 b(Matthies) h(and) e(A.) g(Sc) n(heel,) p Fq 26 w(T) -6 b(r) l(ans.) 28 b(A) n(mer.) g(Math.) f(So) l(c.) p Fr(,) p Fa 27 w(335) p Fr(,) f(747,) h(\(2002\).) 172 3874 y(22.) 51 b(V.) 26 b(E.) g(Zakharo) n(v,) p Fq 26 w(Appl.) h(Me) l(ch.) g(T) -6 b(e) l(ch.) 28 b(Physics) p Fa 27 w(2) p Fr(,) e(190,) h(\(1968\).) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF