Content-Type: multipart/mixed; boundary="-------------0405141644887" This is a multi-part message in MIME format. ---------------0405141644887 Content-Type: text/plain; name="04-153.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-153.keywords" explicit solutions, solitons, solitary waves, ground state, nonlinear scalar field equations. ---------------0405141644887 Content-Type: application/postscript; name="pground.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pground.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: pground.dvi %%Pages: 4 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips pground -o pground.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.05.14:1740 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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b(These)31 b(equations)e(b)s(elong)f(to)0 3885 y(the)36 b(class)g(of)f(nonlinear)g(scalar)g(\014eld)g(equations)h(see)h Fd(e.g.)p Fi(,)f(Berest)m(yc)m(ki)i(&)e(Lions)f(\(1983\))f([1])i(and)0 4006 y(references)i(therein.)53 b(Applications)33 b(include)j(the)g (ground)f(state)h(to)g(the)g(nonlinear)e(Sc)m(hr\177)-49 b(odinger)0 4126 y(equation)48 b(see)h Fd(e.g.)p Fi(,)j(Chaio)c Fd(et)h(al.)90 b Fi(\(1964\))48 b([2])g(and)g(to)g(the)h(Kortew)m (eg{de)g(V)-8 b(ries)48 b(\(1894\))f([3])0 4247 y(equation.)c(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)d(lemma:)0 4415 y Fj(Prop)s(osition)39 b(1.)44 b Fd(F)-7 b(or)37 b(\014xe)-5 b(d)37 b Ff(p)c(>)h Fi(0)p Fd(,)k Ff(c)33 b(>)h Fi(0)j Fd(and)h Ff(\013)c Fg(2)f Fe(R)49 b Fd(the)38 b(e)-5 b(quation)45 b Fi(\(1\))38 b Fd(has)f(solutions)h(that)0 4535 y(de)-5 b(c)g(ay)35 b(to)g(zer)-5 b(o)34 b(as)h Fg(j)p Ff(x)p Fg(j)27 b(!)g(1)35 b Fd(of)f(the)h(form)343 4836 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))29 b(=)683 4666 y Fc( )909 4769 y Ff(\013)p 772 4814 336 4 v 772 4905 a(c)p Fi(\(2)22 b(+)g Ff(p)p Fi(\))1140 4836 y(+)1238 4659 y Fc(s)p 1338 4659 911 4 v 1491 4769 a Fi(1)p 1348 4814 336 4 v 1348 4905 a Ff(c)p Fi(\(1)f(+)h Ff(p)p Fi(\))1715 4836 y(+)1980 4769 y Ff(\013)2043 4740 y Fm(2)p 1823 4814 415 4 v 1823 4905 a Ff(c)1865 4876 y Fm(2)1905 4905 y Fi(\(2)g(+)g Ff(p)p Fi(\))2199 4876 y Fm(2)2265 4836 y Fi(cosh)2466 4756 y Fc(\000)2512 4836 y Ff(p)2561 4760 y Fg(p)p 2644 4760 42 4 v 76 x Ff(c)p Fi(\()p Ff(x)g Fg(\000)h Ff(x)2956 4851 y Fm(0)2996 4836 y Fi(\))3034 4756 y Fc(\001)3079 4666 y(!)3158 4677 y Fb(\000)p Fm(1)p Fl(=p)3340 4836 y Ff(;)219 b Fi(\(2\))0 5097 y Fd(for)35 b(any)f(tr)-5 b(anslation)35 b(c)-5 b(onstant)34 b Ff(x)1275 5112 y Fm(0)1343 5097 y Fg(2)28 b Fe(R)5 b Fd(.)1831 5346 y Fi(1)p eop %%Page: 2 2 2 1 bop 0 289 a Fd(Pr)-5 b(o)g(of.)48 b Fi(Equation)35 b(\(1\))g(is)f(translational)e(in)m(v)-5 b(arian)m(t,)35 b(and)g(hence)h(it)e(su\016ce)j(to)e(consider)g(the)g(case)0 409 y Ff(x)55 424 y Fm(0)128 409 y Fi(=)f(0.)54 b(T)-8 b(o)36 b(v)m(erify)g(that)g(\(2\))g(is)f(a)h(solution)f(to)g(\(1\))h(w) m(e)h(substitute)g(it)e(in)m(to)g(\(1\).)54 b(First)35 b(consider)0 529 y(the)e(term)f Ff(@)457 493 y Fm(2)452 554 y Fl(x)497 529 y Ff(Q)p Fi(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)742 839 y Ff(@)793 854 y Fl(x)837 839 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))28 b(=)g Fg(\000)1254 661 y Fc(s)p 1354 661 754 4 v 1449 772 a Fi(1)p 1364 816 218 4 v 1364 907 a(1)22 b(+)g Ff(p)1614 839 y Fi(+)1859 772 y Ff(\013)1922 743 y Fm(2)p 1722 816 376 4 v 1722 907 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))2058 879 y Fm(2)2124 839 y Fi(sinh\()p Ff(p)2385 762 y Fg(p)p 2468 762 42 4 v 77 x Ff(cx)p Fi(\))p Ff(Q)2680 798 y Fl(p)p Fm(+1)2810 839 y Fi(\()p Ff(x)p Fi(\))p Ff(;)618 b Fi(\(3\))0 1125 y(and)33 b(hence)105 1434 y Fg(\000)23 b Ff(@)261 1393 y Fm(2)256 1459 y Fl(x)301 1434 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))28 b(=)g Ff(p)690 1358 y Fg(p)p 773 1358 V 76 x Ff(c)815 1257 y Fc(s)p 915 1257 754 4 v 1009 1367 a Fi(1)p 925 1412 218 4 v 925 1503 a(1)21 b(+)h Ff(p)1174 1434 y Fi(+)1419 1367 y Ff(\013)1482 1338 y Fm(2)p 1282 1412 376 4 v 1282 1503 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))1618 1474 y Fm(2)1684 1434 y Fi(cosh)q(\()p Ff(p)1956 1358 y Fg(p)p 2039 1358 42 4 v 76 x Ff(cx)p Fi(\))p Ff(Q)2251 1393 y Fl(p)p Fm(+1)2381 1434 y Fi(\()p Ff(x)p Fi(\))1736 1731 y Fg(\000)1835 1591 y Fc(\022)1908 1731 y Fi(1)g(+)2087 1664 y Ff(\013)2150 1628 y Fm(2)2190 1664 y Fi(\()p Ff(p)g Fi(+)g(1\))p 2087 1708 396 4 v 2097 1800 a Ff(c)p Fi(\(2)h(+)f Ff(p)p Fi(\))2434 1771 y Fm(2)2493 1591 y Fc(\023)2583 1731 y Fi(sinh)2757 1688 y Fm(2)2797 1731 y Fi(\()p Ff(p)2884 1655 y Fg(p)p 2966 1655 42 4 v 2966 1731 a Ff(cx)p Fi(\))p Ff(Q)3178 1690 y Fm(2)p Fl(p)p Fm(+1)3344 1731 y Fi(\()p Ff(x)p Fi(\))p Ff(:)84 b Fi(\(4\))0 2004 y(T)-8 b(o)34 b(the)f(end)i(of)e (comparing)f Ff(@)1145 1968 y Fm(2)1140 2029 y Fl(x)1185 2004 y Ff(Q)i Fi(with)f(the)h(remaining)d(terms)i(in)g(\(1\))g(w)m(e)i (break)f(out)f Ff(Q)3354 1968 y Fm(2)p Fl(p)p Fm(+1)3553 2004 y Fi(and)0 2124 y(use)h(the)f(explicit)e(form)g(of)h Ff(Q)1099 2088 y Fl(p)1172 2124 y Fi(to)g(obtain)105 2454 y Fg(\000)23 b Ff(@)261 2413 y Fm(2)256 2478 y Fl(x)301 2454 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))28 b(=)g Ff(Q)718 2413 y Fm(2)p Fl(p)p Fm(+1)883 2454 y Fi(\()p Ff(x)p Fi(\))1031 2283 y Fc( )1335 2386 y Ff(\013)p 1199 2431 336 4 v 1199 2522 a(c)p Fi(\(2)22 b(+)g Ff(p)p Fi(\))1567 2454 y(+)1665 2276 y Fc(s)p 1764 2276 911 4 v 1918 2386 a Fi(1)p 1774 2431 336 4 v 1774 2522 a Ff(c)p Fi(\(1)g(+)g Ff(p)p Fi(\))2142 2454 y(+)2406 2386 y Ff(\013)2469 2357 y Fm(2)p 2250 2431 415 4 v 2250 2522 a Ff(c)2292 2493 y Fm(2)2332 2522 y Fi(\(2)f(+)h Ff(p)p Fi(\))2625 2493 y Fm(2)2691 2454 y Fi(cosh)q(\()p Ff(p)2963 2377 y Fg(p)p 3046 2377 42 4 v 77 x Ff(cx)p Fi(\))3181 2283 y Fc(!)3277 2454 y Fg(\001)455 2797 y Ff(p)504 2720 y Fg(p)p 587 2720 V 77 x Ff(c)629 2619 y Fc(s)p 729 2619 754 4 v 823 2730 a Fi(1)p 739 2774 218 4 v 739 2865 a(1)f(+)h Ff(p)989 2797 y Fi(+)1233 2730 y Ff(\013)1296 2701 y Fm(2)p 1096 2774 376 4 v 1096 2865 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))1432 2837 y Fm(2)1498 2797 y Fi(cosh)q(\()p Ff(p)1770 2720 y Fg(p)p 1853 2720 42 4 v 77 x Ff(cx)p Fi(\))37 b Fg(\000)2119 2657 y Fc(\022)2192 2797 y Fi(1)22 b(+)2371 2730 y Ff(\013)2434 2694 y Fm(2)2473 2730 y Fi(\()p Ff(p)g Fi(+)g(1\))p 2371 2774 396 4 v 2381 2865 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))2717 2837 y Fm(2)2777 2657 y Fc(\023)2867 2797 y Fi(sinh)3041 2754 y Fm(2)3080 2797 y Fi(\()p Ff(p)3167 2720 y Fg(p)p 3250 2720 42 4 v 77 x Ff(cx)p Fi(\))3385 2657 y Fc(\023)3476 2797 y Ff(:)83 b Fi(\(5\))0 3099 y(Recalling)33 b(that)h(cosh)825 3056 y Fm(2)865 3099 y Fi(\()p Ff(y)t Fi(\))22 b Fg(\000)j Fi(sinh)1291 3056 y Fm(2)1330 3099 y Fi(\()p Ff(y)t Fi(\))31 b(=)h(1)j(and)g(collecting)e(equal)i(p)s(o)m(w)m(ers)h(of)f(cosh\()p Fg(\001)p Fi(\))g(together,)0 3220 y(giv)m(es)105 3549 y Fg(\000)23 b Ff(@)261 3508 y Fm(2)256 3573 y Fl(x)301 3549 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))28 b(=)g Ff(Q)718 3508 y Fm(2)p Fl(p)p Fm(+1)883 3549 y Fi(\()p Ff(x)p 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2888 4553 42 4 v 76 x Ff(cx)p Fi(\))3023 4459 y Fc(!)3102 4470 y Fm(2)1051 4973 y Fg(\000)h Ff(\013)1230 4802 y Fc( )1455 4905 y Ff(\013)p 1319 4950 336 4 v 1319 5041 a(c)p Fi(\(2)f(+)g Ff(p)p Fi(\))1687 4973 y(+)1785 4795 y Fc(s)p 1884 4795 911 4 v 2038 4905 a Fi(1)p 1894 4950 336 4 v 1894 5041 a Ff(c)p Fi(\(1)g(+)g Ff(p)p Fi(\))2262 4973 y(+)2526 4905 y Ff(\013)2589 4876 y Fm(2)p 2370 4950 415 4 v 2370 5041 a Ff(c)2412 5012 y Fm(2)2451 5041 y Fi(\(2)g(+)g Ff(p)p Fi(\))2745 5012 y Fm(2)2811 4973 y Fi(cosh)q(\()p Ff(p)3083 4896 y Fg(p)p 3166 4896 42 4 v 77 x Ff(cx)p Fi(\))3301 4802 y Fc(!)17 b(!)3476 4973 y Ff(:)83 b Fi(\(7\))1831 5346 y(2)p eop %%Page: 3 3 3 2 bop 0 289 a Fi(Expanding)33 b(the)g(square)h(and)e(collecting)f (terms)h(of)g(equal)h(p)s(o)m(w)m(ers)h(in)e(cosh\()p Fg(\001)p Fi(\),)h(w)m(e)g(\014nd)83 615 y Ff(cQ)p Fi(\()p Ff(x)p Fi(\))23 b Fg(\000)f Ff(Q)532 573 y Fm(2)p Fl(p)p Fm(+1)698 615 y Fi(\()p Ff(x)p Fi(\))g Fg(\000)h Ff(\013)q(Q)1091 573 y Fl(p)p Fm(+1)1221 615 y Fi(\()p Ff(x)p Fi(\))28 b(=)f Ff(Q)1560 573 y Fm(2)p Fl(p)p Fm(+1)1726 615 y Fi(\()p Ff(x)p Fi(\))1857 444 y Fc( )1936 474 y(\022)2104 547 y Fi(1)p 2019 592 218 4 v 2019 683 a(1)22 b(+)g Ff(p)2269 615 y Fi(+)2513 547 y Ff(\013)2576 511 y Fm(2)p 2377 592 376 4 v 2377 683 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))2713 654 y Fm(2)2762 474 y Fc(\023)2852 615 y Fi(cosh)3037 572 y Fm(2)3076 615 y Fi(\()p Ff(p)3163 538 y Fg(p)p 3246 538 42 4 v 77 x Ff(cx)p Fi(\))791 958 y Fg(\000)1054 891 y Ff(p\013)p 900 935 419 4 v 900 955 a Fg(p)p 983 955 42 4 v 72 x Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))1329 780 y Fc(s)p 1429 780 754 4 v 1523 891 a Fi(1)p 1439 935 218 4 v 1439 1026 a(1)g(+)g Ff(p)1689 958 y Fi(+)1933 891 y Ff(\013)1996 862 y Fm(2)p 1797 935 376 4 v 1797 1026 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))2133 998 y Fm(2)2198 958 y Fi(cosh)q(\()p Ff(p)2470 881 y Fg(p)p 2553 881 42 4 v 77 x Ff(cx)p Fi(\))h Fg(\000)f Fi(1)g Fg(\000)2991 891 y Ff(\013)3054 854 y Fm(2)3093 891 y Fi(\(1)g(+)g Ff(p)p Fi(\))p 2991 935 396 4 v 3001 1026 a Ff(c)p Fi(\(2)g(+)g Ff(p)p Fi(\))3337 998 y Fm(2)3397 788 y Fc(!)3476 958 y Ff(:)83 b Fi(\(8\))0 1259 y(Since)33 b(\(6\))f(is)g(min)m(us)g (\(8\),)g(summing)f(them)h(yields)h(zero.)p 3640 1259 4 66 v 3644 1196 59 4 v 3644 1259 V 3702 1259 4 66 v 0 1457 a Fj(Remark)g(1.)38 b Fd(F)-7 b(or)31 b Ff(\013)d Fi(=)g(0)j Fd(we)g(r)-5 b(e)g(c)g(over)31 b(the)g(wel)5 b(l)31 b(known)g(solution,)g(se)-5 b(e)31 b(von)g(Sz.-Nagy)g(\(1941\))g ([4],)0 1577 y(Titchmarsh)j(\(1946\))g([5])941 1812 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))28 b(=)1281 1732 y Fc(\000)1326 1812 y Ff(c)p Fi(\(1)22 b(+)g Ff(p)p Fi(\))1662 1732 y Fc(\001)1708 1754 y Fm(1)p Fl(=)p Fm(2)p Fl(p)1853 1812 y Fi(sec)m(h)2030 1769 y Fm(1)p Fl(=p)2140 1732 y Fc(\000)2186 1812 y Ff(p)2235 1736 y Fg(p)p 2318 1736 42 4 v 76 x Ff(c)p Fi(\()p Ff(x)g Fi(+)g Ff(m)p Fi(\))2696 1732 y Fc(\001)2742 1812 y Ff(:)817 b Fi(\(9\))0 2026 y Fj(Remark)37 b(2.)42 b Fd(The)34 b(class)g(of)h(e)-5 b(quations)1098 2240 y Fg(\000)p Ff(@)1231 2199 y Fm(2)1226 2265 y Fl(x)1272 2240 y Ff(Q)22 b Fi(+)g Ff(cQ)h Fg(\000)f Ff(\014)6 b(Q)1848 2199 y Fm(2)p Fl(p)p Fm(+1)2036 2240 y Fg(\000)22 b Ff(\013)q(Q)2275 2199 y Fl(p)p Fm(+1)2433 2240 y Fi(=)27 b(0)p Ff(;)925 b Fi(\(10\))0 2454 y Fd(with)31 b Ff(\014)i(>)28 b Fi(0)p Fd(,)k Ff(c)27 b(>)h Fi(0)p Fd(,)j(and)g Ff(\013)d Fg(2)g Fe(R)43 b Fd(have)30 b(solutions,)i(that) f(de)-5 b(c)g(ay)31 b(to)h(zer)-5 b(o)30 b(as)h Fg(j)p Ff(x)p Fg(j)d(!)f(1)p Fd(,)32 b(of)f(the)g(form)257 2790 y Ff(Q)p Fi(\()p Ff(x)p Fi(\))d(=)597 2619 y Fc( )822 2722 y Ff(\013)p 685 2767 336 4 v 685 2858 a(c)p Fi(\(2)22 b(+)g Ff(p)p Fi(\))1053 2790 y(+)1151 2612 y Fc(s)p 1251 2612 911 4 v 1398 2722 a Ff(\014)p 1261 2767 336 4 v 1261 2858 a(c)p Fi(\(1)g(+)g Ff(p)p Fi(\))1629 2790 y(+)1893 2722 y Ff(\013)1956 2693 y Fm(2)p 1737 2767 415 4 v 1737 2858 a Ff(c)1779 2829 y Fm(2)1818 2858 y Fi(\(2)g(+)g Ff(p)p Fi(\))2112 2829 y Fm(2)2178 2790 y Fi(cosh)2379 2709 y Fc(\000)2425 2790 y Ff(p)2474 2713 y Fg(p)p 2557 2713 42 4 v 77 x Ff(c)p Fi(\()p Ff(x)g Fg(\000)h Ff(x)2869 2805 y Fm(0)2909 2790 y Fi(\))2947 2709 y Fc(\001)2992 2619 y(!)3071 2630 y Fb(\000)p Fm(1)p Fl(=p)3253 2790 y Ff(;)257 b Fi(\(11\))0 3085 y Fd(for)40 b(any)g(tr)-5 b(anslation)40 b(c)-5 b(onstant)40 b Ff(x)1297 3100 y Fm(0)1337 3085 y Fd(.)61 b(This)40 b(r)-5 b(esult)41 b(fol)5 b(lows)39 b(dir)-5 b(e)g(ctly)41 b(fr)-5 b(om)39 b(Pr)-5 b(op)g(osition)40 b(1)g(as)g(the)0 3205 y(r)-5 b(esc)g(aling)46 b(tr)-5 b(ansformation)47 b Fg(f)p Ff(\013)q(;)17 b(c;)g(x)30 b Fg(\000)i Ff(x)1582 3220 y Fm(0)1622 3205 y Fg(g)50 b(7!)h(f)p Ff(\013)q(\014)6 b(;)17 b(c\014)6 b(;)17 b Fi(\()p Ff(x)30 b Fg(\000)h Ff(x)2524 3220 y Fm(0)2564 3205 y Fi(\))p Ff(\014)2663 3169 y Fb(\000)p Fm(1)p Fl(=)p Fm(2)2828 3205 y Fg(g)47 b Fd(maps)54 b Fi(\(10\))47 b Fd(to)54 b Fi(\(1\))o Fd(.)0 3326 y(F)-7 b(urthermor)i(e,)40 b(in)f(the)h(limit)f Ff(\014)j Fg(!)36 b Fi(0)p Fd(,)41 b Ff(\013)c(>)f Fi(0)p Fd(,)k(using)g(the)f (`half-angle)f(formula')h(for)h Fi(cosh)q(\()p Fg(\001)p Fi(\))f Fd(we)0 3446 y(onc)-5 b(e)34 b(again)g(r)-5 b(e)g(c)g(over)34 b(the)h(solution)42 b Fi(\(9\))p Fd(,)35 b(with)g Ff(p)27 b Fg(7!)g Ff(p=)p Fi(2)p Fd(.)0 3644 y Fj(Remark)54 b(3.)49 b Fd(F)-7 b(or)48 b(the)g(nonline)-5 b(ar)48 b(eigenvalue)f(p)-5 b(ar)g(ameter,)51 b Ff(c)p Fd(,)h(the)d(solution)f(is)g(a)h(one)f(bump) 0 3765 y(solution)35 b(for)f(al)5 b(l)35 b(p)-5 b(ositive)34 b(values)g(of)h Ff(c)p Fd(.)45 b(Thus)34 b(ther)-5 b(e)35 b(ar)-5 b(e)35 b(no)f(excite)-5 b(d)34 b(states.)0 3963 y Fj(Remark)h(4.)40 b Fd(Consider)32 b(the)i(de)-5 b(c)g(aying-to-zer)g (o)32 b(at)h(in\014nity)g(solutions)g(to)h(the)f(class)g(of)g(e)-5 b(quations)1247 4188 y Fg(\000)p Ff(@)1380 4147 y Fm(2)1375 4213 y Fl(x)1420 4188 y Ff(Q)23 b Fi(+)f Ff(cQ)g Fg(\000)1859 4094 y Fc(X)1873 4305 y Fl(j)t Fb(2)p Fl(I)2019 4188 y Ff(a)2070 4203 y Fl(j)2107 4188 y Ff(Q)2184 4147 y Fl(p)2220 4157 y Fa(j)2284 4188 y Fi(=)28 b(0)p Ff(;)1073 b Fi(\(12\))0 4505 y Fd(for)39 b(c)-5 b(onstants)38 b Fg(f)p Ff(a)693 4520 y Fl(j)729 4505 y Ff(;)17 b(p)822 4520 y Fl(j)858 4505 y Fg(g)908 4520 y Fl(j)t Fb(2)p Fl(I)1028 4505 y Fd(,)39 b(with)g Ff(a)1364 4520 y Fl(j)1435 4505 y Fg(2)c Fe(R)50 b Fd(and)38 b Ff(p)1889 4520 y Fl(j)1960 4505 y Ff(>)d Fi(0)k Fd(wher)-5 b(e)38 b Ff(I)k Fg(\032)36 b Fe(Z)p Fd(.)53 b(That)39 b(e)-5 b(quation)45 b Fi(\(12\))39 b Fd(is)0 4625 y(tr)-5 b(anslational)34 b(invariant)h(suggest)g(the)g(change)f(of)g(variable)g Ff(v)e Fi(=)c Ff(@)2520 4640 y Fl(x)2565 4625 y Ff(Q)p Fd(,)35 b Ff(@)2758 4640 y Fl(x)2802 4625 y Ff(v)d Fi(=)c Ff(@)3036 4640 y Fl(Q)3096 4625 y Ff(v)20 b(@)3214 4640 y Fl(x)3259 4625 y Ff(Q)28 b Fi(=)g Ff(v)t(@)3570 4640 y Fl(Q)3630 4625 y Ff(v)t Fd(.)0 4745 y(R)-5 b(eplac)g(e)34 b Ff(@)411 4709 y Fm(2)406 4770 y Fl(x)451 4745 y Ff(Q)h Fd(in)g(terms)g(of)f Ff(v)39 b Fd(yields)34 b(a)h(sep)-5 b(ar)g(able)34 b(e)-5 b(quation,)34 b(inte)-5 b(gr)g(ation)35 b(gives)386 5007 y Fi(2)435 4966 y Fb(\000)p Fm(1)529 5007 y Ff(v)580 4966 y Fm(2)647 5007 y Fi(=)751 4872 y Fc(Z)867 5007 y Ff(cQ)22 b Fg(\000)1108 4913 y Fc(X)1122 5124 y Fl(j)t Fb(2)p Fl(I)1269 5007 y Ff(a)1320 5022 y Fl(j)1356 5007 y Ff(Q)1433 4966 y Fl(p)1469 4976 y Fa(j)1541 5007 y Fi(d)p Ff(Q)28 b Fi(=)f(2)1852 4966 y Fb(\000)p Fm(1)1946 5007 y Ff(cQ)2065 4966 y Fm(2)2127 5007 y Fg(\000)2227 4913 y Fc(X)2241 5124 y Fl(j)t Fb(2)p Fl(I)2371 5007 y Fi(\()p Ff(p)2458 5022 y Fl(j)2516 5007 y Fi(+)22 b(1\))2701 4966 y Fb(\000)p Fm(2)2795 5007 y Ff(a)2846 5022 y Fl(j)2883 5007 y Ff(Q)2960 4966 y Fl(p)2996 4976 y Fa(j)3028 4966 y Fm(+1)3145 5007 y Fi(+)g Ff(k)s(;)213 b Fi(\(13\))1831 5346 y(3)p eop %%Page: 4 4 4 3 bop 0 289 a Fd(for)36 b(some)g(c)-5 b(onstant)36 b Ff(k)s Fd(.)49 b(F)-7 b(or)36 b(functions)g(that)h(de)-5 b(c)g(ay)36 b(at)h(in\014nity)f(b)-5 b(oth)36 b Ff(Q)h Fd(and)f Ff(v)e Fi(=)c Ff(@)3214 304 y Fl(x)3259 289 y Ff(Q)36 b Fd(ar)-5 b(e)37 b(zer)-5 b(o)0 409 y(at)31 b(in\014nity,)g(thus)h Ff(k)e Fi(=)e(0)p Fd(.)43 b(When)31 b(the)g(solution,)g Ff(Q)p Fd(,)h(exists)e(and)g(is)h(smo)-5 b(oth,)31 b(the)g(critic)-5 b(al)30 b(p)-5 b(oints)31 b(of)0 529 y Ff(Q)37 b Fd(ar)-5 b(e)36 b(at)h Ff(@)451 544 y Fl(x)495 529 y Ff(Q)31 b Fi(=)g(0)36 b Fd(and)46 b Fi(\(13\))36 b Fd(gives)g(the)h(explicit)f(value)g(of)g Ff(Q)h Fd(at)f(these)h(p)-5 b(oints)36 b(as)g(the)h(solution)0 650 y(to)e(the)g(p)-5 b(olynomial)34 b(e)-5 b(quation)1177 882 y Fi(2)1226 841 y Fb(\000)p Fm(1)1320 882 y Ff(cQ)1439 841 y Fm(2)1507 882 y Fi(=)1610 787 y Fc(X)1625 999 y Fl(j)t Fb(2)p Fl(I)1754 882 y Fi(\()p Ff(p)1841 897 y Fl(j)1900 882 y Fi(+)22 b(1\))2085 841 y Fb(\000)p Fm(2)2179 882 y Ff(a)2230 897 y Fl(j)2266 882 y Ff(Q)2343 841 y Fl(p)2379 851 y Fa(j)2412 841 y Fm(+1)2506 882 y Ff(:)1004 b Fi(\(14\))0 1199 y Fd(This)38 b(has)h(applic)-5 b(ations)38 b(as)g(the)h(starting)h(p)-5 b(oint)38 b(for)h(a)g(sho)-5 b(oting)38 b(algorithm,)i(when)e(numeric)-5 b(al)5 b(ly)0 1319 y(solving)43 b Fi(\(13\))o Fd(.)0 1652 y Fh(References)0 1871 y Fi([1])49 b(Berest)m(yc)m(ki,)32 b(H.)e(&)g(Lions,)g(P)-8 b(.-L.)29 b(1983)39 b(Nonlinear)28 b(Scalar)h(Field)g(Equations.)h(I.)g (Existence)i(of)103 1991 y(a)g(ground)h(state.)44 b Fd(A)n(r)-5 b(ch.)34 b(R)-5 b(ational)34 b(Me)-5 b(ch.)35 b(A)n(nal.)p Fi(,)d Fj(82)p Fi(\(4\),)g(313{345.)0 2195 y([2])49 b(Chiao,)36 b(R.)g(Y.,)i(Garmire,)d(E.)i(&)g(T)-8 b(o)m(wnes,)38 b(C.)f(H.)g(1964)55 b(Self-T)-8 b(rapping)35 b(of)h(Optical)f(Beams.) 103 2315 y Fd(Phys.)g(R)-5 b(ev.)34 b(L)-5 b(ett.)33 b Fj(13)p Fi(,)g(479{482.)0 2518 y([3])49 b(Kortew)m(eg,)22 b(D.)g(F.)f(&)h(de)g(V)-8 b(ries,)22 b(G.)f(1894)k(On)c(the)i(Change)f (of)f(F)-8 b(orm)21 b(of)g(Long)g(W)-8 b(a)m(v)m(es)24 b(adv)-5 b(ancing)103 2639 y(in)29 b(a)h(Rectangular)f(Canal)g(and)h (on)g(a)f(New)i(T)m(yp)s(e)g(of)f(Long)f(Stationary)g(W)-8 b(a)m(v)m(es.)41 b(Philosophical)103 2759 y(Magazine,)32 b(5th)h(series,)g Fj(36)p Fi(,)g(422{443.)0 2962 y([4])49 b(v)m(on)26 b(Sz.-Nagy)-8 b(,)27 b(B.)f(1941)1116 2937 y(\177)1103 2962 y(Ub)s(er)h(In)m(tegralungleic)m(h)m(unger)e(zwisc)m (hen)j(einer)e(F)-8 b(unktion)25 b(und)i(ihrer)103 3083 y(Ableitung.)42 b Fd(A)-5 b(cta)36 b(Sci.)e(Math.)h(\(Sze)-5 b(ge)g(d\))p Fi(,)31 b Fj(10)p Fi(,)i(64{74.)0 3286 y([5])49 b(Titc)m(hmarsh,)35 b(E.)g(C.)g(1946)49 b Fd(Eigenfunction)36 b(exp)-5 b(ansions)36 b(asso)-5 b(ciate)g(d)36 b(with)g(se)-5 b(c)g(ond-or)g(der)36 b(dif-)103 3407 y(fer)-5 b(ential)34 b(e)-5 b(quations.)35 b(Part)g(1.)43 b Fi(Oxford:)g(Clarendon)33 b(Press.)1831 5346 y(4)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0405141644887--