Content-Type: multipart/mixed; boundary="-------------0004200312884" This is a multi-part message in MIME format. ---------------0004200312884 Content-Type: text/plain; name="00-191.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-191.keywords" generalized Kadomtsev-Petviashvili equations, traveling waves ---------------0004200312884 Content-Type: application/postscript; name="kp2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kp2.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.74 Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: kp_2.dvi %%CreationDate: Thu Sep 2 18:55:09 1999 %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: C:\EDITORS\EMTEX\BIN\DVIPS.EXE -o kp_2.ps kp_2.dvi %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 1999.09.02:1853 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B 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Fo(\)\))1219 1567 y Fl(x)1254 1560 y Fo(=)14 b(0)g Fm(:)363 b Fo(\(6\))164 1667 y(Solitary)21 b(w)o(a)o(v)o(es)g(are)g(solutions)h (of)g(the)f(same)g(equation)g(\(6\),)i(with)e Fm(k)k Fo(=)d Fj(1)p Fo(.)37 b(The)164 1727 y(action)16 b(functional)g(asso)q (ciated)h(with)g(\(6\))f(reads)h([17])678 1858 y Fm(J)705 1865 y Fl(k)727 1858 y Fo(\()p Fm(u)p Fo(\))c(=)863 1824 y(1)p 863 1847 25 2 v 863 1892 a(2)892 1858 y Fj(k)p Fm(u)p Fj(k)970 1838 y Fi(2)970 1870 y Fl(k)1002 1858 y Fj(\000)1052 1790 y Fk(Z)1080 1903 y Fl(Q)1108 1909 y Fg(k)1137 1858 y Fm(F)7 b Fo(\()p Fm(u)p Fo(\))13 b(;)452 b(\(7\))164 2000 y Fm(J)191 2007 y Fl(k)230 2000 y Fo(is)18 b(of)g(the)g(class)g Fm(C)579 1982 y Fi(1)617 2000 y Fo(on)g Fm(X)726 2007 y Fl(k)748 2000 y Fo(.)27 b(W)l(e)17 b(consider)h(w)o(eak)g(solutions)g(of)h(\(6\),)f(i.e.)26 b(critical)164 2060 y(p)q(oin)o(ts)17 b(of)f Fm(J)393 2067 y Fl(k)431 2060 y Fo(in)g Fm(X)528 2067 y Fl(k)549 2060 y Fo(.)237 2120 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b(with)f(minimal)d(action)k(among)f(all)164 436 y(non)o(trivial)15 b(solutions,)h(solving)h(the)f(follo)o(wing)g (minim)o(iz)o(ation)e(problem)693 526 y Fm(m)736 533 y Fl(k)771 526 y Fo(=)g(inf)s Fj(f)p Fm(J)934 533 y Fl(k)955 526 y Fo(\()p Fm(u)p Fo(\))f(:)h Fm(u)f Fj(2)h Fm(S)1180 533 y Fl(k)1202 526 y Fj(g)f Fm(:)467 b Fo(\(9\))164 615 y(Remark)15 b(that)554 687 y Fm(J)581 694 y Fl(k)603 687 y Fo(\()p Fm(u)p Fo(\))e(=)734 620 y Fk(Z)762 732 y Fl(Q)790 738 y Fg(k)824 654 y Fo(1)p 824 676 25 2 v 824 721 a(2)853 687 y Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))11 b Fj(\000)g Fm(F)c Fo(\()p Fm(u)p Fo(\))13 b Fm(;)56 b(u)14 b Fj(2)g Fm(S)1344 694 y Fl(k)1379 687 y Fm(:)303 b Fo(\(10\))164 796 y(In)22 b(what)h(follo)o(ws)f(w)o(e)g(will)f(omit)g (the)h(subscript)g Fm(k)j Fo(if)c Fm(k)26 b Fo(=)e Fj(1)f Fo(and)f(write)g(simply)164 856 y Fm(J)o(;)8 b(I)t(;)g(:)g(:)g(:)237 916 y Fo(Throughout)23 b(this)d(section,)h(in)g(addition)g(to)g (Assumptions)f(1)1435 898 y Fi(0)1455 916 y Fo({4)1503 898 y Fi(0)1523 916 y Fo(,)i(w)o(e)e(imp)q(ose)164 976 y(the)c(follo)o(wing)g(one)164 1061 y Fp(\(N\))24 b Fe(F)l(or)17 b(any)g Fm(u)d Fj(2)g Fm(L)582 1043 y Fi(2)602 1061 y Fo(\()p Ff(R)660 1043 y Fi(2)676 1061 y Fo(\))k Fe(such)g(that)875 1107 y Fk(Z)902 1220 y Fd(R)929 1210 y Fc(2)954 1175 y Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))14 b Fm(>)f Fo(0)i Fm(;)286 1288 y Fe(the)j(function)h(of)e Fm(t)888 1369 y(t)906 1349 y Fh(\000)p Fi(1)961 1301 y Fk(Z)989 1414 y Fd(R)1016 1405 y Fc(2)1040 1369 y Fm(uf)5 b Fo(\()p Fm(tu)p Fo(\))286 1474 y Fe(is)17 b(strictly)h(incr)n(e)n(asing)g(on)f Fo(\(0)p Fm(;)8 b Fo(+)p Fj(1)p Fo(\).)237 1558 y(In)18 b(the)h(pro)q(of)g(of)g(Theorem)e(1,)i([17)q(],)f(w)o(e)g(ha)o(v)o(e)g (considered)g(the)g(Moun)o(tain)g(P)o(ass)164 1619 y(V)l(alues)13 b Fm(c)337 1626 y Fl(k)372 1619 y Fo(for)g Fm(J)470 1626 y Fl(k)505 1619 y Fo(and)h(pro)o(v)o(ed)f(that)h(they)e(are)i (uniformly)d(b)q(ounded)k(from)d(b)q(elo)o(w)h(and)164 1679 y(ab)q(o)o(v)o(e)j(b)o(y)g(p)q(ositiv)o(e)f(constan)o(ts.)22 b(More)16 b(precisely)l(,)713 1769 y Fm(c)734 1776 y Fl(k)769 1769 y Fo(=)34 b(inf)821 1802 y Fl(\015)r Fh(2)p Fi(\000)887 1808 y Fg(k)906 1802 y Fi(\))933 1769 y Fo(max)928 1802 y Fl(t)p Fh(2)p Fi([0)p Fl(;)p Fi(1])1037 1769 y Fm(J)1064 1776 y Fl(k)1085 1769 y Fo(\()p Fm(\015)s Fo(\()p Fm(t)p Fo(\)\))13 b Fm(;)164 1883 y Fo(where)448 1943 y(\000)478 1950 y Fl(k)513 1943 y Fo(=)h Fj(f)p Fm(\015)j Fj(2)d Fm(C)t Fo(\([0)p Fm(;)8 b Fo(1])p Fm(;)g(X)897 1950 y Fl(k)918 1943 y Fo(\))14 b(:)f Fm(\015)s Fo(\(0\))h(=)g(0)p Fm(;)8 b(J)1207 1950 y Fl(k)1229 1943 y Fo(\()p Fm(\015)s Fo(\(1\)\))14 b Fm(<)f Fo(0)p Fj(g)i Fm(:)164 2022 y Fo(Here)h(w)o(e)h(ha)o(v)o(e)f(de\014ned)h(\000)665 2029 y Fl(k)704 2022 y Fo(in)g(a)h(sligh)o(tly)e(di\013eren)o(t)g(w)o(a)o(y) h(than)h(in)f([17],)g(but)g(it)g(do)q(es)164 2082 y(not)g(e\013ect)e (on)i(the)f(v)m(alue)g(of)h Fm(c)734 2089 y Fl(k)755 2082 y Fo(.)k(Consider)c(also)g(another)f(minimax)d(v)m(alue)747 2172 y Fm(c)768 2151 y Fh(0)768 2184 y Fl(k)803 2172 y Fo(=)34 b(inf)855 2211 y Fl(v)q Fh(2)p Fl(X)929 2196 y Fc(+)926 2223 y Fg(k)962 2172 y Fo(sup)970 2211 y Fl(t>)p Fi(0)1044 2172 y Fm(J)1071 2179 y Fl(k)1092 2172 y Fo(\()p Fm(tv)r Fo(\))13 b Fm(;)164 2296 y Fo(where)632 2368 y Fm(X)676 2347 y Fi(+)672 2381 y Fl(k)720 2368 y Fo(=)h Fj(f)p Fm(v)h Fj(2)f Fm(X)923 2375 y Fl(k)959 2368 y Fo(:)986 2300 y Fk(Z)1014 2413 y Fl(Q)1042 2419 y Fg(k)1071 2368 y Fm(F)7 b Fo(\()p Fm(v)r Fo(\))12 b Fm(>)i Fo(0)p Fj(g)g Fm(:)164 2482 y Fo(Due)i(to)h(Assumption)e(4)620 2464 y Fi(0)640 2482 y Fo(,)h Fm(X)714 2462 y Fi(+)710 2496 y Fl(k)758 2482 y Fj(6)p Fo(=)d Fj(;)p Fo(.)961 2607 y(4)p eop %%Page: 5 6 5 5 bop 164 315 a Fp(Lemma)16 b(1)24 b Fe(F)l(or)f(every)h Fm(v)i Fj(2)g Fm(X)800 295 y Fi(+)796 329 y Fl(k)853 315 y Fe(ther)n(e)d(exists)i(a)e(unique)i Fm(t)1349 322 y Fl(k)1395 315 y Fo(=)g Fm(t)1476 322 y Fl(k)1497 315 y Fo(\()p Fm(v)r Fo(\))e Fe(such)h(that)164 376 y Fm(t)182 383 y Fl(k)203 376 y Fm(v)15 b Fj(2)f Fm(S)319 383 y Fl(k)340 376 y Fe(,)738 436 y Fm(J)765 443 y Fl(k)786 436 y Fo(\()p Fm(t)823 443 y Fl(k)844 436 y Fm(v)r Fo(\))f(=)h(max)970 466 y Fl(t>)p Fi(0)1053 436 y Fm(J)1080 443 y Fl(k)1101 436 y Fo(\()p Fm(tv)r Fo(\))f Fm(;)164 545 y Fe(and)18 b Fm(t)277 552 y Fl(k)298 545 y Fo(\()p Fm(v)r Fo(\))e Fe(dep)n(ends)i(c)n(ontinuously)h(on)e Fm(v)f Fj(2)e Fm(X)1042 524 y Fi(+)1038 559 y Fl(k)1071 545 y Fe(.)237 659 y(Pr)n(o)n(of)9 b Fo(.)21 b(Assumption)15 b(4)690 641 y Fi(0)727 659 y Fo(implies)e(that)827 723 y Fk(Z)855 835 y Fl(Q)883 841 y Fg(k)912 790 y Fm(v)r(f)5 b Fo(\()p Fm(v)r Fo(\))13 b Fm(>)g Fo(0)164 937 y(for)j(an)o(y)h Fm(v)e Fj(2)f Fm(X)461 916 y Fi(+)457 951 y Fl(k)491 937 y Fo(.)21 b(Therefore,)15 b(due)i(to)f(Assumption)f Fp(\(N\))p Fo(,)g(the)h(function)506 1039 y Fm(d)p 497 1061 43 2 v 497 1107 a(dt)545 1073 y(J)572 1080 y Fl(k)593 1073 y Fo(\()p Fm(tv)r Fo(\))d(=)g Fm(I)761 1080 y Fl(k)782 1073 y Fo(\()p Fm(tv)r Fo(\))g(=)h Fm(t)947 1052 y Fi(2)966 1073 y Fo(\()p Fj(k)p Fm(v)r Fj(k)1061 1052 y Fi(2)1061 1085 y Fl(k)1093 1073 y Fj(\000)d Fm(t)1161 1052 y Fh(\000)p Fi(1)1216 1005 y Fk(Z)1243 1118 y Fl(Q)1271 1124 y Fg(k)1300 1073 y Fm(v)r(f)5 b Fo(\()p Fm(tv)r Fo(\)\))164 1218 y(v)m(anishes)15 b(at)h(only)f(one)g(p)q(oin)o(t)g Fm(t)751 1225 y Fl(k)786 1218 y Fo(=)f Fm(t)856 1225 y Fl(k)877 1218 y Fo(\()p Fm(v)r Fo(\))f Fm(>)h Fo(0.)21 b(Equation)15 b(\(10\))h(and)g(Assumption)e(4)1763 1200 y Fi(0)164 1278 y Fo(imply)g(that)j Fm(J)434 1285 y Fl(k)471 1278 y Fo(is)g(p)q(ositiv)o(e)e(on)i Fm(S)799 1285 y Fl(k)820 1278 y Fo(.)22 b(Since)15 b Fm(J)1010 1285 y Fl(k)1032 1278 y Fo(\(0\))f(=)g(0,)i(w)o(e)g(see)g(that)h Fm(t)1489 1285 y Fl(k)1526 1278 y Fo(is)f(a)h(p)q(oin)o(t)g(of)164 1338 y(maxim)o(um)12 b(for)k Fm(J)493 1345 y Fl(k)514 1338 y Fo(\()p Fm(tv)r Fo(\).)21 b(Con)o(tin)o(uit)o(y)14 b(of)j Fm(t)948 1345 y Fl(k)969 1338 y Fo(\()p Fm(v)r Fo(\))e(is)i(easy)f(to)g(v)o(erify)l(.)k Fb(2)164 1452 y Fp(Lemma)c(2)24 b Fm(c)436 1459 y Fl(k)471 1452 y Fo(=)14 b Fm(c)544 1434 y Fh(0)544 1465 y Fl(k)579 1452 y Fo(=)g Fm(m)674 1459 y Fl(k)695 1452 y Fe(.)237 1566 y(Pr)n(o)n(of)9 b Fo(.)26 b(Since)17 b Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))17 b(is)h(sub)q(quadratic)g(at)g(0)g(and)h(the)e(quadratic)h(part)g(of)g Fm(J)1711 1573 y Fl(k)1750 1566 y Fo(is)164 1627 y(p)q(ositiv)o(e)c (de\014ned,)g(w)o(e)f(see)h(that)h Fm(I)795 1634 y Fl(k)816 1627 y Fo(\()p Fm(v)r Fo(\))e Fm(>)h Fo(0)h(in)f(a)g(neigh)o(b)q(orho)q (o)q(d)j(of)d(the)h(origin,)f(exept)164 1687 y(of)19 b(0.)28 b(Hence,)17 b Fm(I)471 1694 y Fl(k)492 1687 y Fo(\()p Fm(\015)s Fo(\()p Fm(t)p Fo(\)\))g Fm(>)g Fo(0,)i Fm(\015)i Fj(2)c Fo(\000)869 1694 y Fl(k)891 1687 y Fo(,)i(for)f(small) f Fm(t)g(>)h Fo(0.)28 b(Due)18 b(to)h(Assumption)e(4)1749 1669 y Fi(0)1769 1687 y Fo(,)164 1747 y(for)f Fm(v)g Fj(2)e Fm(X)369 1726 y Fi(+)365 1761 y Fl(k)415 1747 y Fo(w)o(e)i(ha)o(v)o(e)434 1883 y(2)p Fm(J)485 1890 y Fl(k)507 1883 y Fo(\()p Fm(v)r Fo(\))d(=)h Fj(k)p Fm(v)r Fj(k)712 1862 y Fi(2)712 1895 y Fl(k)744 1883 y Fj(\000)c Fo(2)825 1815 y Fk(Z)854 1928 y Fl(Q)882 1934 y Fg(k)911 1883 y Fm(F)d Fo(\()p Fm(v)r Fo(\))12 b Fm(>)i Fj(k)p Fm(v)r Fj(k)1154 1862 y Fi(2)1154 1895 y Fl(k)1186 1883 y Fj(\000)d Fm(\026)1273 1815 y Fk(Z)1301 1928 y Fl(Q)1329 1934 y Fg(k)1358 1883 y Fm(F)c Fo(\()p Fm(v)r Fo(\))13 b Fj(\025)651 2056 y(\025)h(k)p Fm(v)r Fj(k)780 2035 y Fi(2)780 2068 y Fl(k)811 2056 y Fj(\000)d Fo(2)893 1988 y Fk(Z)921 2101 y Fl(Q)949 2107 y Fg(k)979 2056 y Fm(v)r(f)5 b Fo(\()p Fm(v)r Fo(\))12 b(=)i Fm(I)1184 2063 y Fl(k)1205 2056 y Fo(\()p Fm(v)r Fo(\))f Fm(:)164 2176 y Fo(Hence,)d Fm(I)340 2183 y Fl(k)361 2176 y Fo(\()p Fm(\015)s Fo(\(1\)\))k Fm(<)g Fo(0.)20 b(Therefore,)11 b Fm(\015)s Fo(\()p Fm(t)p Fo(\))g(crosses)h Fm(S)1124 2183 y Fl(k)1156 2176 y Fo(and)g(this)f(implies)d(that)k Fm(c)1618 2183 y Fl(k)1653 2176 y Fj(\025)h Fm(m)1748 2183 y Fl(k)1769 2176 y Fo(.)237 2236 y(By)f(Assumption)f(4)600 2218 y Fi(0)620 2236 y Fo(,)i(for)g(an)o(y)g Fm(v)i Fj(2)f Fm(X)937 2215 y Fi(+)933 2250 y Fl(k)979 2236 y Fo(w)o(e)e(ha)o(v)o(e)g Fm(F)7 b Fo(\()p Fm(tv)r Fo(\))12 b Fj(\025)i Fm(\013t)1391 2218 y Fl(\026)1414 2236 y Fo(,)f Fm(\013)h(>)g Fo(0,)f(if)f Fm(t)h(>)h Fo(0)f(is)164 2296 y(large)i(enough.)22 b(This)16 b(implies)c(that)k Fm(J)876 2303 y Fl(k)897 2296 y Fo(\()p Fm(tv)r Fo(\))d Fm(<)h Fo(0)i(for)g(ev)o(ery)d Fm(v)j Fj(2)e Fm(X)1416 2275 y Fi(+)1412 2310 y Fl(k)1461 2296 y Fo(and)i(su\016cien)o(tly)164 2356 y(large)h Fm(t)f(>)g Fo(0.)25 b(Hence,)16 b(the)h(half-axis)h Fj(f)p Fm(tv)e Fo(:)g Fm(t)f(>)h Fo(0)p Fj(g)i Fo(generates)f(in)h(a)f(natural)h(w)o (a)o(y)f(an)164 2416 y(elemen)o(t)c(of)k(\000)430 2423 y Fl(k)452 2416 y Fo(.)k(This)16 b(implies)e(the)i(inequalit)o(y)e Fm(c)1095 2423 y Fl(k)1130 2416 y Fj(\024)g Fm(c)1204 2398 y Fh(0)1204 2429 y Fl(k)1225 2416 y Fo(.)961 2607 y(5)p eop %%Page: 6 7 6 6 bop 237 315 a Fo(No)o(w)16 b(let)g Fm(v)f Fj(2)f Fm(S)535 322 y Fl(k)556 315 y Fo(.)21 b(By)16 b(the)g(de\014nition)g (of)g Fm(I)1046 322 y Fl(k)1067 315 y Fo(,)766 447 y Fm(\033)f Fo(=)861 379 y Fk(Z)889 492 y Fl(Q)917 498 y Fg(k)946 447 y Fm(v)r(f)5 b Fo(\()p Fm(v)r Fo(\))13 b Fm(>)g Fo(0)i Fm(;)164 590 y Fo(and)i Fp(\(N\))e Fo(implies)f(that) 439 691 y Fm(d)p 430 713 43 2 v 430 759 a(dt)486 657 y Fk(Z)514 769 y Fl(Q)542 775 y Fg(k)571 724 y Fm(F)7 b Fo(\()p Fm(tv)r Fo(\))12 b(=)756 657 y Fk(Z)783 769 y Fl(Q)811 775 y Fg(k)841 724 y Fm(v)r(f)5 b Fo(\()p Fm(tv)r Fo(\))12 b Fj(\025)i Fm(t)1061 704 y Fh(\000)p Fi(1)1116 657 y Fk(Z)1143 769 y Fl(Q)1171 775 y Fg(k)1201 724 y Fm(v)r(f)5 b Fo(\()p Fm(tv)r Fo(\))12 b Fj(\025)i Fm(\033)h(>)f Fo(0)164 864 y(pro)o(vided)h Fm(t)f Fj(\025)g Fo(1.)21 b(Hence,)15 b(for)h Fm(t)e(>)f Fo(0)k(large)f(enough)813 928 y Fk(Z)840 1041 y Fl(Q)868 1047 y Fg(k)898 996 y Fm(F)7 b Fo(\()p Fm(tv)r Fo(\))12 b Fm(>)h Fo(0)i Fm(:)164 1137 y Fo(By)h(de\014nitions)g(of)g Fm(c)553 1119 y Fh(0)553 1150 y Fl(k)591 1137 y Fo(and)g Fm(m)728 1144 y Fl(k)749 1137 y Fo(,)g(w)o(e)g(see)g(that)h Fm(c)1057 1119 y Fh(0)1057 1150 y Fl(k)1092 1137 y Fo(=)d Fm(m)1187 1144 y Fl(k)1208 1137 y Fo(.)21 b Fb(2)164 1252 y Fp(Theorem)c(1)24 b Fe(Assume)15 b Fo(1)657 1234 y Fi(0)677 1252 y Fe({)p Fo(4)726 1234 y Fi(0)760 1252 y Fe(and)g Fp(\(N\))f Fe(to)h(b)n(e)f (ful\014le)n(d.)23 b(Then,)16 b(for)e(any)g Fm(k)i Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(\))p Fe(,)164 1312 y(ther)n(e)16 b(exists)h(a)f(minimizer)g Fm(u)716 1319 y Fl(k)751 1312 y Fj(2)e Fm(S)828 1319 y Fl(k)866 1312 y Fe(of)i(\(9\))g(which)g(is)g (a)h(critic)n(al)f(p)n(oint)g(of)g Fm(J)1598 1319 y Fl(k)1619 1312 y Fe(.)22 b(Mor)n(e-)164 1372 y(over,)i Fm(J)320 1379 y Fl(k)341 1372 y Fo(\()p Fm(u)388 1379 y Fl(k)409 1372 y Fo(\))f(=)g Fm(m)555 1379 y Fl(k)598 1372 y Fe(is)g(b)n(ounde)n (d)f(fr)n(om)f(ab)n(ove)i(and)g(b)n(elow)g(by)g(p)n(ositive)f(c)n (onstants)164 1432 y(indep)n(endent)d(on)f Fm(k)r Fe(.)237 1546 y(Pr)n(o)n(of)9 b Fo(.)32 b(In)19 b(the)h(pro)q(of)h(of)f(Theorem) e(1,)j([17],)f(it)f(is)h(sho)o(wn)g(that)g(there)f(exists)h(a)164 1607 y(P)o(alais-Smale)15 b(sequence)g Fm(u)681 1614 y Fl(k)q(;n)747 1607 y Fj(2)f Fm(X)834 1614 y Fl(k)872 1607 y Fo(at)j(the)f(lev)o(el)e Fm(c)1148 1614 y Fl(k)1169 1607 y Fo(,)i(i.e.)654 1717 y Fm(J)686 1696 y Fh(0)681 1729 y Fl(k)702 1717 y Fo(\()p Fm(u)749 1724 y Fl(k)q(;n)801 1717 y Fo(\))e Fj(!)g Fo(0)g Fm(;)57 b(J)1034 1724 y Fl(k)1055 1717 y Fo(\()p Fm(u)1102 1724 y Fl(k)q(;n)1154 1717 y Fo(\))14 b Fj(!)g Fm(c)1272 1724 y Fl(k)164 1827 y Fo(as)22 b Fm(n)g Fj(!)g(1)p Fo(.)36 b(Moreo)o(v)o(er,)21 b Fm(u)714 1834 y Fl(k)q(;n)789 1827 y Fj(!)h Fm(u)889 1834 y Fl(k)931 1827 y Fo(w)o(eakly)e(in)h Fm(X)1199 1834 y Fl(k)1242 1827 y Fo(and)h(strongly)f(in)g Fm(L)1630 1803 y Fl(p)1630 1840 y(loc)1676 1827 y Fo(\()p Ff(R)1734 1808 y Fi(2)1750 1827 y Fo(\),)164 1887 y(where)16 b Fm(u)333 1894 y Fl(k)368 1887 y Fj(2)e Fm(X)455 1894 y Fl(k)493 1887 y Fo(is)i(a)h(non)o(trivial)e(solution)h(of)h(\(6\).)k (Therefore,)655 1997 y Fm(I)677 2004 y Fl(k)698 1997 y Fo(\()p Fm(u)745 2004 y Fl(k)q(;n)798 1997 y Fo(\))13 b(=)h Fj(h)p Fm(J)933 1976 y Fh(0)928 2009 y Fl(k)950 1997 y Fo(\()p Fm(u)997 2004 y Fl(k)q(;n)1049 1997 y Fo(\))p Fm(;)8 b(u)1118 2004 y Fl(k)q(;n)1170 1997 y Fj(i)14 b(!)g Fo(0)164 2107 y(and)358 2179 y Fm(J)385 2186 y Fl(k)407 2179 y Fo(\()p Fm(u)454 2186 y Fl(k)q(;n)506 2179 y Fo(\))d Fj(\000)591 2145 y Fo(1)p 591 2167 25 2 v 591 2213 a(2)620 2179 y Fm(I)642 2186 y Fl(k)663 2179 y Fo(\()p Fm(u)710 2186 y Fl(k)q(;n)762 2179 y Fo(\))j(=)847 2111 y Fk(Z)875 2224 y Fl(Q)903 2230 y Fg(k)924 2179 y Fo(\()948 2145 y(1)p 948 2167 V 948 2213 a(2)977 2179 y Fm(u)1005 2186 y Fl(k)q(;n)1057 2179 y Fm(f)5 b Fo(\()p Fm(u)1133 2186 y Fl(k)q(;n)1186 2179 y Fo(\))11 b Fj(\000)g Fm(F)c Fo(\()p Fm(u)1352 2186 y Fl(k)q(;n)1403 2179 y Fo(\)\))14 b Fj(!)g Fm(c)1540 2186 y Fl(k)1575 2179 y Fm(:)164 2301 y Fo(Due)f(to)g(4)343 2283 y Fi(0)363 2301 y Fo(,)h(the)e(in)o(tegrand)h(here)f(is)h(nonnegativ)o(e)g(and,)h (since)e Fm(u)1352 2308 y Fl(k)q(;n)1418 2301 y Fj(!)i Fm(u)1510 2308 y Fl(k)1544 2301 y Fo(in)e Fm(L)1630 2283 y Fi(2)1630 2314 y Fl(loc)1676 2301 y Fo(\()p Ff(R)1734 2283 y Fi(2)1750 2301 y Fo(\),)164 2361 y(w)o(e)k(ha)o(v)o(e)655 2365 y Fk(Z)683 2478 y Fl(Q)711 2484 y Fg(k)732 2433 y Fo(\()756 2399 y(1)p 756 2421 V 756 2467 a(2)785 2433 y Fm(u)813 2440 y Fl(k)834 2433 y Fm(f)5 b Fo(\()p Fm(u)910 2440 y Fl(k)931 2433 y Fo(\))12 b Fj(\000)e Fm(F)d Fo(\()p Fm(u)1097 2440 y Fl(k)1118 2433 y Fo(\)\))13 b Fj(\024)h Fm(c)1243 2440 y Fl(k)1278 2433 y Fm(:)961 2607 y Fo(6)p eop %%Page: 7 8 7 7 bop 164 315 a Fo(Ho)o(w)o(ev)o(er,)17 b Fm(u)405 322 y Fl(k)444 315 y Fo(is)h(a)h(non)o(trivial)f(solution,)g(hence,)g Fm(u)1143 322 y Fl(k)1182 315 y Fj(2)f Fm(S)1262 322 y Fl(k)1284 315 y Fo(.)28 b(Therefore,)18 b(w)o(e)g(deduce)164 376 y(from)d(\(10\))i(that)544 510 y Fm(J)571 517 y Fl(k)592 510 y Fo(\()p Fm(u)639 517 y Fl(k)660 510 y Fo(\))d(=)745 442 y Fk(Z)772 555 y Fl(Q)800 561 y Fg(k)821 510 y Fo(\()845 476 y(1)p 845 498 25 2 v 845 544 a(2)875 510 y Fm(u)903 517 y Fl(k)924 510 y Fm(f)5 b Fo(\()p Fm(u)1000 517 y Fl(k)1021 510 y Fo(\))11 b Fj(\000)g Fm(F)c Fo(\()p Fm(u)1187 517 y Fl(k)1207 510 y Fo(\)\))14 b Fj(\025)g Fm(m)1355 517 y Fl(k)1390 510 y Fm(:)164 653 y Fo(No)o(w)i(Lemma)e(2)j(implies)c (that)k Fm(J)788 660 y Fl(k)809 653 y Fo(\()p Fm(u)856 660 y Fl(k)877 653 y Fo(\))d(=)g Fm(m)1005 660 y Fl(k)1042 653 y Fo(and)j Fm(u)1165 660 y Fl(k)1202 653 y Fo(is)f(a)h(ground)g(w)o (a)o(v)o(e)f(solution.)237 713 y(The)21 b(last)g(statemen)o(t)e(of)i (the)g(theorem)e(follo)o(ws)i(imme)o(diately)c(from)j(Lemma)f(2)164 773 y(and)e(uniform)e(estimates)f(for)j Fm(c)752 780 y Fl(k)773 773 y Fo(.)k Fb(2)164 887 y Fp(Remark)16 b(1)25 b Fo(The)15 b(Nehari)g(v)m(ariational)h(principle)e(suggested)i(in)f ([13)q(])g(w)o(as)h(used)g(suc-)164 947 y(cessfully)j(in)h(man)o(y)g (pap)q(ers)h(\(see,)g(e.g.,)f([2,)g(9)q(,)g(14)q(,)g(15,)h(16,)f(17)q (,)g(22)q(]\).)34 b(In)20 b(all)g(these)164 1008 y(pap)q(ers,)15 b(exept)e(of)i([16],)f(the)g(geometry)e(of)j(Nehari)e(manifold)g(is)h (simple)e(enough:)20 b(it)14 b(is)164 1068 y(a)k(b)q(ounded)h(surface)f (without)h(b)q(oundary)g(around)g(the)f(origin,)g(lik)o(e)e(sphere.)26 b(In)18 b(the)164 1128 y(case)13 b(w)o(e)f(consider)g(here)g(the)h (picture)f(is)g(di\013eren)o(t:)19 b Fm(S)1148 1135 y Fl(k)1182 1128 y Fo(ma)o(y)11 b(lo)q(ok)o(e)h(lik)o(e)f(sphere)i(if,)f (e.g.,)164 1188 y Fm(f)5 b Fo(\()p Fm(u)p Fo(\))14 b(=)h Fj(j)p Fm(u)p Fj(j)382 1170 y Fl(p)p Fh(\000)p Fi(2)446 1188 y Fm(u)p Fo(,)h(and)h(ma)o(y)e(b)q(e)i(un)o(b)q(ounded)g(if,)f (e.g.,)f Fm(f)5 b Fo(\()p Fm(u)p Fo(\))15 b(=)f Fj(j)p Fm(u)p Fj(j)1404 1170 y Fl(p)p Fh(\000)p Fi(1)1468 1188 y Fo(.)23 b(Nev)o(ertheless,)164 1248 y(in)16 b(an)o(y)g(case)h Fm(S)447 1255 y Fl(k)484 1248 y Fo(separates)g(the)g(origin)f(and)h (the)f(domain)g(of)g(negativ)o(e)g(v)m(alues)g(of)h Fm(J)1748 1255 y Fl(k)1769 1248 y Fo(,)164 1309 y(whic)o(h)c(is)h(su\016cien)o(t) f(for)h(our)h(purp)q(ose.)21 b(In)14 b([16)q(])f(suc)o(h)h(a)h (manifold)e(is)g(also)i(un)o(b)q(ounded)164 1369 y(in)h(general,)f(but) i(there)e(w)o(e)h(ha)o(v)o(e)g(used)g(di\013eren)o(t)g(argumen)o(ts.) 237 1483 y(No)o(w)k(w)o(e)g(are)g(going)i(to)e(study)h(the)f(b)q(eha)o (vior)g(of)h Fm(u)1229 1490 y Fl(k)1250 1483 y Fo(,)g(as)g Fm(k)h Fj(!)f(1)p Fo(.)33 b(Recall)19 b(the)164 1543 y(de\014nition)i(of)g(cut-o\013)h(op)q(erators)g Fm(P)855 1550 y Fl(k)899 1543 y Fo(:)f Fm(X)974 1550 y Fl(k)1018 1543 y Fj(!)h Fm(X)t Fo(,)g([17].)35 b(Let)22 b Fm(\037)1419 1550 y Fl(k)1461 1543 y Fj(2)h Fm(C)1556 1525 y Fh(1)1552 1555 y Fi(0)1592 1543 y Fo(\()p Ff(R)p Fo(\))18 b(b)q(e)k(a)164 1603 y(nonnegativ)o(e)16 b(function)g(suc)o(h)g(that)h Fm(\037)873 1610 y Fl(k)894 1603 y Fo(\()p Fm(x)p Fo(\))d(=)g(1)j(for)g Fm(x)c Fj(2)i Fo([)p Fj(\000)p Fm(k)r(=)p Fo(2)p Fm(;)8 b(k)r(=)p Fo(2],)16 b Fm(\037)1531 1610 y Fl(k)1552 1603 y Fo(\()p Fm(x)p Fo(\))e(=)g(0)j(for)164 1663 y Fj(j)p Fm(x)p Fj(j)c(\025)h Fo(\()p Fm(k)f Fo(+)e(1\))p Fm(=)p Fo(2,)17 b(and)g Fj(j)p Fm(\037)654 1645 y Fh(0)654 1676 y Fl(k)674 1663 y Fj(j)p Fm(;)8 b Fj(j)p Fm(\037)755 1645 y Fh(00)755 1676 y Fl(k)776 1663 y Fj(j)13 b(\024)h Fm(C)891 1670 y Fi(0)927 1663 y Fo(,)i(with)g(some)f(constan)o(t)i Fm(C)1422 1670 y Fi(0)1455 1663 y Fm(>)d Fo(0.)22 b(W)l(e)16 b(set)631 1773 y Fm(P)662 1780 y Fl(k)683 1773 y Fm(u)p Fo(\()p Fm(x;)8 b(y)r Fo(\))13 b(=)h([)p Fm(\037)935 1780 y Fl(k)955 1773 y Fo(\()p Fm(x)p Fo(\))p Fm(D)1062 1753 y Fh(\000)p Fi(1)1061 1787 y Fl(x;k)1112 1773 y Fm(u)p Fo(\()p Fm(x;)8 b(y)r Fo(\)])1268 1780 y Fl(x)1303 1773 y Fm(:)164 1888 y Fp(Theorem)17 b(2)24 b Fe(Assume)f(that)g Fo(1)770 1870 y Fi(0)790 1888 y Fe({)p Fo(4)839 1870 y Fi(0)882 1888 y Fe(and)g Fo(\(N\))f Fe(ar)n(e)g(satis\014e)n(d.)39 b(L)n(et)22 b Fm(u)1506 1895 y Fl(k)1551 1888 y Fj(2)i Fm(X)1648 1895 y Fl(k)1693 1888 y Fe(b)n(e)f(a)164 1948 y(se)n(quenc)n(e)16 b(of)e(gr)n(ound)g(wave)i(solutions.)22 b(Then)15 b(ther)n(e)f(exist)h(a)f(nontrivial)i(gr)n(ound)e(wave)164 2008 y Fm(u)21 b Fj(2)g Fm(X)26 b Fe(and)c(a)f(se)n(quenc)n(e)i(of)e (ve)n(ctors)h Fm(\020)930 2015 y Fl(k)973 2008 y Fj(2)g Ff(R)1067 1990 y Fi(2)1105 2008 y Fe(such)f(that,)i(along)g(a)e(subse)n (quenc)n(e,)164 2068 y Fm(P)195 2075 y Fl(k)217 2068 y Fm(u)245 2075 y Fl(k)266 2068 y Fo(\()p Fj(\001)11 b Fo(+)g Fm(\020)380 2075 y Fl(k)401 2068 y Fo(\))j Fj(!)g Fm(u)j Fe(we)n(akly)h(in)g Fm(X)t Fe(.)23 b(If)17 b(in)h(addition)396 2178 y Fj(j)p Fm(f)5 b Fo(\()p Fm(u)11 b Fo(+)g Fm(v)r Fo(\))f Fj(\000)h Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fj(j)13 b(\024)h Fm(C)t Fo(\(1)d(+)g Fj(j)p Fm(u)p Fj(j)1024 2158 y Fl(p)p Fh(\000)p Fi(2)1099 2178 y Fo(+)g Fj(j)p Fm(v)r Fj(j)1202 2158 y Fl(p)p Fh(\000)p Fi(2)1266 2178 y Fo(\))p Fj(j)p Fm(v)r Fj(j)i Fm(;)50 b(v)15 b Fj(2)f Ff(R)p Fm(;)141 b Fo(\(11\))164 2288 y Fe(then,)18 b(along)h(the)f (same)f(subse)n(quenc)n(e,)702 2398 y Fo(lim)691 2430 y Fl(k)q Fh(!1)789 2398 y Fj(k)p Fm(u)842 2405 y Fl(k)863 2398 y Fo(\()p Fj(\001)11 b Fo(+)g Fm(\020)977 2405 y Fl(k)999 2398 y Fo(\))g Fj(\000)f Fm(u)p Fj(k)1131 2405 y Fl(k)1166 2398 y Fo(=)k(0)p Fm(:)961 2607 y Fo(7)p eop %%Page: 8 9 8 8 bop 237 315 a Fe(Pr)n(o)n(of)9 b Fo(.)20 b(By)14 b(Theorem)e(2,)j([17],)e(there)h(is)f(a)i(non)o(trivial)e(solution)h Fm(u)f Fj(2)h Fm(X)19 b Fo(suc)o(h)13 b(that)164 376 y Fm(P)195 383 y Fl(k)217 376 y Fm(u)245 383 y Fl(k)266 376 y Fo(\()p Fj(\001)d Fo(+)g Fm(\020)378 383 y Fl(k)400 376 y Fo(\))j Fj(!)h Fm(u)i Fo(w)o(eakly)e(in)i Fm(X)k Fo(for)c(some)f Fm(\020)1034 383 y Fl(k)1069 376 y Fj(2)f Ff(R)1155 358 y Fi(2)1188 376 y Fo(\(alonge)i(a)g(subsequence\).)21 b(Let)164 436 y(us)c(pro)o(v)o(e)e(that)i Fm(u)f Fo(is)g(a)g(ground)i (w)o(a)o(v)o(e,)c(i.e.)642 540 y Fm(J)5 b Fo(\()p Fm(u)p Fo(\))13 b(=)h(inf)s Fj(f)p Fm(J)5 b Fo(\()p Fm(v)r Fo(\))13 b(:)g Fm(v)i Fj(2)f Fm(S)s Fj(g)g Fo(=)g Fm(m)f(:)237 643 y Fo(First)j(of)g(all,)f(for)h(an)o(y)g Fm(v)f Fj(2)f Fm(S)19 b Fo(and)e(an)o(y)f Fm(")d(>)h Fo(0,)i(there)g(exist)f Fm(k)1387 650 y Fl(")1421 643 y Fo(and)i Fm(v)1540 650 y Fl(k)1575 643 y Fj(2)d Fm(S)1652 650 y Fl(k)1689 643 y Fo(suc)o(h)164 704 y(that)675 764 y Fm(J)702 771 y Fl(k)723 764 y Fo(\()p Fm(v)766 771 y Fl(k)787 764 y Fo(\))g Fj(\024)g Fm(J)5 b Fo(\()p Fm(v)r Fo(\))10 b(+)h Fm(";)56 b(k)16 b Fj(\025)d Fm(k)1239 771 y Fl(")1258 764 y Fm(:)164 848 y Fo(Indeed,)18 b(since)g Fm(J)24 b Fo(and)19 b Fm(I)k Fo(are)c(con)o(tin)o(uous,)g(w)o(e)f(can)h(\014nd) g Fm(')1297 855 y Fl(k)1337 848 y Fj(2)f Fm(C)1427 830 y Fh(1)1423 861 y Fi(0)1464 848 y Fo(\()p Fm(Q)1522 855 y Fl(k)1543 848 y Fo(\))h(suc)o(h)f(that)164 909 y Fm(\021)188 916 y Fl(k)223 909 y Fo(=)c Fm(D)315 916 y Fl(x)337 909 y Fm(')369 916 y Fl(k)405 909 y Fj(!)f Fm(v)18 b Fo(in)e Fm(X)k Fo(and,)d(hence,)605 1012 y Fm(J)5 b Fo(\()p Fm(\021)680 1019 y Fl(k)701 1012 y Fo(\))14 b Fj(!)g Fm(J)5 b Fo(\()p Fm(v)r Fo(\))p Fm(;)55 b(I)t Fo(\()p Fm(\021)1032 1019 y Fl(k)1053 1012 y Fo(\))14 b Fj(!)f Fm(I)t Fo(\()p Fm(v)r Fo(\))g(=)h(0)p Fm(:)164 1116 y Fo(Since)h Fm(I)t Fo(\()p Fm(v)r Fo(\))e(=)h(0)i(and)h Fm(v)e Fj(6)p Fo(=)f(0,)i(w)o(e)g(ha)o(v)o (e)740 1176 y Fk(Z)768 1289 y Fl(Q)796 1295 y Fg(k)825 1244 y Fm(v)r(f)5 b Fo(\()p Fm(v)r Fo(\))13 b(=)g Fj(k)p Fm(v)r Fj(k)1084 1224 y Fi(2)1117 1244 y Fm(>)h Fo(0)p Fm(:)164 1378 y Fo(Hence,)807 1391 y Fk(Z)835 1504 y Fl(Q)863 1510 y Fg(k)892 1459 y Fm(\021)916 1466 y Fl(k)937 1459 y Fm(f)5 b Fo(\()p Fm(\021)1009 1466 y Fl(k)1031 1459 y Fo(\))14 b Fm(>)g Fo(0)164 1577 y(for)g Fm(k)i Fo(large)d(enough.)22 b(Due)13 b(to)h(\(N\),)f(there)g(exists)g Fm(\034)1113 1584 y Fl(k)1149 1577 y Fm(>)h Fo(0)g(suc)o(h)f(that)h Fm(I)t Fo(\()p Fm(\034)1515 1584 y Fl(k)1536 1577 y Fm(\021)1560 1584 y Fl(k)1582 1577 y Fo(\))f(=)h(0)g(and)164 1637 y Fm(\034)185 1644 y Fl(k)220 1637 y Fj(!)g Fo(1.)21 b(Let)16 b Fm(v)454 1644 y Fl(k)490 1637 y Fo(b)q(e)g(a)g(unique)f Fm(k)r Fo(-p)q(erio)q(dic)g(function)g(whic)o(h)g(coincides)g(with)g Fm(\034)1649 1644 y Fl(k)1671 1637 y Fm(\021)1695 1644 y Fl(k)1731 1637 y Fo(on)164 1697 y Fm(Q)203 1704 y Fl(k)224 1697 y Fo(.)21 b(Then)674 1757 y Fm(J)701 1764 y Fl(k)723 1757 y Fo(\()p Fm(v)766 1764 y Fl(k)786 1757 y Fo(\))14 b(=)g Fm(J)5 b Fo(\()p Fm(\034)943 1764 y Fl(k)964 1757 y Fm(\021)988 1764 y Fl(k)1009 1757 y Fo(\))14 b Fj(\024)g Fm(J)5 b Fo(\()p Fm(v)r Fo(\))10 b(+)h Fm(")164 1842 y Fo(pro)o(vided)k Fm(k)k Fo(is)d(large)g(enough.)237 1902 y(In)g(particular,)g(w)o(e)f(ha)o(v)o(e)793 2006 y(lim)f(sup)869 2047 y Fl(k)q Fh(!1)967 2006 y Fm(m)1010 2013 y Fl(k)1045 2006 y Fj(\024)g Fm(m:)164 2137 y Fo(No)o(w,)i (exactly)f(as)h(in)g(the)g(pro)q(of)i(of)e(Theorem)f(5,)h([17)q(],)f(w) o(e)h(see)g(that)711 2241 y(lim)22 b(inf)787 2272 y Fl(k)q Fh(!1)885 2241 y Fm(m)928 2248 y Fl(k)963 2241 y Fj(\025)14 b Fm(J)5 b Fo(\()p Fm(u)p Fo(\))13 b Fj(\025)g Fm(m:)164 2362 y Fo(Hence,)h Fm(m)g Fo(=)g Fm(J)5 b Fo(\()p Fm(u)p Fo(\))15 b(and)i Fm(u)f Fo(is)g(a)h(ground)g(w)o(a)o(v)o(e)e(solution.) 237 2422 y(The)h(second)h(part)g(of)f(the)g(theorem)f(follo)o(ws)h (from)f(Theorem)g(3,)i([17],)e(exactly)g(as)164 2482 y(at)i(the)f(end)g(of)g(pro)q(of)i(of)e(Theorem)f(5,)h([17)q(].)k Fb(2)961 2607 y Fo(8)p eop %%Page: 9 10 9 9 bop 164 315 a Fn(3)81 b(Qualitativ)n(e)25 b(Prop)r(erties)i(of)g(T) -7 b(ra)n(v)n(eling)25 b(W)-7 b(a)n(v)n(es)164 425 y Fo(No)o(w)15 b(w)o(e)f(are)h(going)h(to)g(study)f(suc)o(h)g(prop)q (erties)g(of)g(KP)g(tra)o(v)o(eling)f(w)o(a)o(v)o(es)g(as)i(symme-)164 485 y(try)l(,)f(regularit)o(y)h(and)g(deca)o(y)l(.)21 b(W)l(e)16 b(start)h(with)f(the)g(follo)o(wing)164 584 y Fp(Lemma)g(3)24 b Fe(Supp)n(ose)e(that)h(Assumptions)f Fo(1)1026 566 y Fi(0)1069 584 y Fe(and)g Fo(2)1192 566 y Fi(0)1234 584 y Fe(ar)n(e)g(satis\014e)n(d.)37 b(In)22 b(the)h(c)n(ase)164 644 y Fo(2)14 b Fm(<)g(p)g Fj(\024)g Fo(5)p Fe(,)k(assume)f(in)h(addition)g(that)f Fm(f)i Fj(2)14 b Fm(C)1051 626 y Fi(2)1071 644 y Fo(\()p Ff(R)p Fo(\))g Fe(and)477 741 y Fj(j)p Fm(f)520 721 y Fi(\()p Fl(j)r Fi(\))566 741 y Fo(\()p Fm(u)p Fo(\))p Fj(j)f(\024)h Fm(C)t Fo(\(1)c(+)h Fj(j)p Fm(u)p Fj(j)909 721 y Fl(p)p Fh(\000)p Fi(1)p Fh(\000)p Fl(j)1017 741 y Fo(\))p Fm(;)58 b(j)17 b Fo(=)d(1)p Fm(;)8 b Fo(2)p Fm(;)58 b(u)14 b Fj(6)p Fo(=)f(0)p Fm(:)226 b Fo(\(12\))164 838 y Fe(Then)20 b(any)f(tr)n(aveling)h(wave)h(is)e(c)n(ontinuous.)28 b(Mor)n(e)n(over,)18 b(any)i(solitary)e(\(r)n(esp.)27 b(p)n(eri-)164 898 y(o)n(dic\))17 b(wave)i(tends)f(to)f(zer)n(o)g(as)g Fo(\()p Fm(x;)8 b(y)r Fo(\))13 b Fj(!)h(1)j Fe(\(r)n(esp.)22 b Fm(y)15 b Fj(!)f(1)p Fe(\).)237 997 y(Pr)n(o)n(of)9 b Fo(.)21 b(F)l(or)c(suc)o(h)f(a)g(w)o(a)o(v)o(e)g Fm(u)d Fj(2)h Fm(X)882 1004 y Fl(k)904 997 y Fo(,)i(w)o(e)g(ha)o(v)o(e)593 1094 y Fj(\000)p Fm(cv)677 1101 y Fl(xx)729 1094 y Fj(\000)11 b Fm(v)803 1101 y Fl(y)q(y)853 1094 y Fo(+)g Fm(v)926 1101 y Fl(xxxx)1021 1094 y Fo(=)j Fm(f)5 b Fo(\()p Fm(u)p Fo(\))1168 1101 y Fl(xx)1223 1094 y Fo(=)14 b Fm(g)1298 1101 y Fl(xx)1340 1094 y Fm(;)342 b Fo(\(13\))164 1191 y(Let)586 1274 y(\()p Fj(F)641 1281 y Fl(k)q(;x)692 1274 y Fm(h)p Fo(\)\()p Fm(\030)r Fo(\))14 b(=)866 1207 y Fk(Z)916 1220 y Fl(k)q(=)p Fi(2)894 1319 y Fh(\000)p Fl(k)q(=)p Fi(2)986 1274 y Fm(h)p Fo(\()p Fm(x)p Fo(\))8 b(exp\()p Fj(\000)p Fm(i\030)r(x)p Fo(\))p Fm(dx)164 1391 y Fo(b)q(e)24 b(the)g(F)l(ourier)f(transform)h(if)f Fm(k)29 b Fo(=)e Fj(1)d Fo(\(then)g(w)o(e)f(write)h(simply)d Fj(F)1532 1398 y Fl(x)1554 1391 y Fo(\),)26 b(and)e(the)164 1451 y(sequence)14 b(of)i(F)l(ourier)f(co)q(e\016cien)o(ts)f(if)h Fm(k)g(<)f Fj(1)p Fo(.)21 b(In)15 b(the)g(last)h(case)f Fm(\030)i Fj(2)d Fo(\(2)p Fm(\031)r(=k)r Fo(\))p Ff(Z)-11 b Fo(.)18 b(No)o(w)164 1511 y(w)o(e)e(get)g(from)f(\(13\))662 1571 y Fj(F)698 1578 y Fl(k)q(;x)749 1571 y Fj(F)785 1578 y Fl(y)805 1571 y Fm(u)f Fo(=)g Fm(p)p Fo(\()p Fm(\030)963 1578 y Fi(1)983 1571 y Fm(;)8 b(\030)1026 1578 y Fi(2)1046 1571 y Fo(\)\()p Fj(F)1120 1578 y Fl(k)q(;x)1171 1571 y Fj(F)1207 1578 y Fl(y)1227 1571 y Fm(g)r Fo(\))p Fm(;)411 b Fo(\(14\))164 1653 y(where)634 1730 y Fm(p)p Fo(\()p Fm(\030)r Fo(\))15 b(=)f Fm(p)p Fo(\()p Fm(\030)850 1737 y Fi(1)870 1730 y Fm(;)8 b(\030)913 1737 y Fi(2)933 1730 y Fo(\))14 b(=)1137 1696 y Fm(\030)1160 1678 y Fi(2)1158 1709 y(1)p 1023 1718 272 2 v 1023 1764 a Fm(c\030)1067 1747 y Fi(2)1065 1776 y(1)1098 1764 y Fo(+)d Fm(\030)1170 1747 y Fi(4)1168 1776 y(1)1202 1764 y Fo(+)g Fm(\030)1274 1747 y Fi(2)1272 1776 y(2)1299 1730 y Fm(;)164 1836 y(\030)185 1843 y Fi(1)227 1836 y Fo(and)22 b Fm(\030)348 1843 y Fi(2)389 1836 y Fo(are)g(dual)f(v)m(ariables)h(to)g Fm(x)f Fo(and)h Fm(y)r Fo(,)g(resp)q(ectiv)o(ely)l(.)34 b(If)21 b Fm(k)k Fo(=)e Fj(1)p Fo(,)f(there)f(is)164 1897 y(nothing)g(to)g(do.) 34 b(In)20 b(the)g(case)g(5)i Fm(<)f(p)g(<)g Fo(6)f(one)h(needs)f(only) g(to)h(rep)q(eat)g(the)f(pro)q(of)164 1957 y(of)i(Theorem)e(1.1,)j ([6],)f(whic)o(h)e(do)q(es)j(not)f(use)f(an)o(y)h(particular)f(prop)q (ert)o(y)g(of)h(p)q(o)o(w)o(er)164 2017 y(nonlinearit)o(y)l(,)16 b(exept)h(of)h(its)f(gro)o(wth)h(rate.)25 b(In)17 b(the)h(case)f(2)f Fm(<)g(p)h Fj(\024)f Fo(5)i(the)f(argumen)o(ts)164 2077 y(from)k(the)h(pro)q(of)h(of)g(Lemma)d(4.1,)j([5],)g(w)o(ork)f(and)h (just)f(here)f(assumption)h(\(12\))h(is)164 2137 y(needed.)237 2197 y(No)o(w)h(w)o(e)f(explain)g(ho)o(w)h(to)g(co)o(v)o(er)e(the)i (case)f(of)h(p)q(erio)q(dic)g(w)o(a)o(v)o(es.)43 b(Recall)22 b(the)164 2258 y(follo)o(wing)e(Lizorkin)g(theorem,)f([11].)33 b(Let)20 b Fm(p)p Fo(\()p Fm(\030)r Fo(\),)i Fm(\030)i Fj(2)d Ff(R)1251 2240 y Fl(n)1271 2258 y Fo(,)g(b)q(e)f(of)h(the)f (class)g Fm(C)1681 2240 y Fl(n)1725 2258 y Fo(for)164 2318 y Fj(j)p Fm(\030)199 2325 y Fl(j)217 2318 y Fj(j)14 b Fm(>)g Fo(0,)i Fm(j)h Fo(=)d(1)p Fm(;)8 b(:)g(:)g(:)f(;)h(n)p Fo(.)22 b(Assume)14 b(that)660 2447 y Fj(j)p Fm(\030)697 2426 y Fl(k)715 2431 y Fc(1)695 2460 y Fi(1)743 2447 y Fj(\001)8 b(\001)g(\001)h Fm(\030)833 2427 y Fl(k)851 2431 y Fg(n)831 2460 y Fl(n)972 2414 y Fm(@)1001 2396 y Fl(k)1022 2414 y Fm(p)p 880 2436 259 2 v 880 2487 a(@)s(\030)932 2465 y Fl(k)950 2470 y Fc(1)930 2499 y Fi(1)978 2487 y Fj(\001)f(\001)g(\001)h Fm(@)s(\030)1097 2472 y Fl(k)1115 2476 y Fg(n)1095 2499 y Fl(n)1144 2447 y Fj(j)k(\024)h Fm(M)r(;)961 2607 y Fo(9)p eop %%Page: 10 11 10 10 bop 164 315 a Fo(with)23 b Fm(k)307 322 y Fl(j)350 315 y Fo(=)i(0)e(or)g(1,)i Fm(k)i Fo(=)e Fm(k)729 322 y Fi(1)764 315 y Fo(+)16 b Fj(\001)8 b(\001)g(\001)g Fm(k)909 322 y Fl(n)958 315 y Fo(=)25 b(0)p Fm(;)8 b Fo(1)p Fm(;)g(:)g(:)g(:)g(;)g(n)p Fo(.)41 b(Then)23 b Fm(p)p Fo(\()p Fm(\030)r Fo(\))h(is)e(a)h(F)l(ourier)164 376 y(m)o(ultiplie)o(r)14 b(on)i Fm(L)488 358 y Fl(r)507 376 y Fo(\()p Ff(R)565 358 y Fl(n)586 376 y Fo(\),)g(1)e Fm(<)g(r)h(<)f Fj(1)p Fo(.)237 436 y(W)l(e)i(rewrite)f(no)o(w)i(\(14\)) g(as)g(follo)o(ws)548 522 y Fj(F)584 529 y Fl(k)q(;x)635 522 y Fm(u)c Fo(=)h Fj(F)769 502 y Fh(\000)p Fi(1)764 534 y Fl(y)816 522 y Fo([)p Fm(p)p Fo(\()p Fm(\030)894 529 y Fi(1)914 522 y Fm(;)8 b(\030)957 529 y Fi(2)977 522 y Fo(\))p Fj(F)1032 529 y Fl(y)1052 522 y Fj(F)1088 529 y Fl(k)q(;x)1139 522 y Fm(g)r Fo(])14 b(=)f Fm(P)7 b Fo(\()p Fm(\030)1321 529 y Fi(1)1341 522 y Fo(\))p Fm(g)r(;)164 608 y Fo(where)23 b Fm(P)7 b Fo(\()p Fm(\030)390 615 y Fi(1)410 608 y Fo(\))23 b(is)g(the)g(op)q(erator)h Fj(F)843 590 y Fh(\000)p Fi(1)838 621 y Fl(y)890 608 y Fm(p)p Fo(\()p Fm(\030)954 615 y Fi(1)974 608 y Fm(;)8 b Fj(\001)p Fo(\))p Fj(F)1065 615 y Fl(y)1108 608 y Fo(for)24 b(an)o(y)f(\014xed)f Fm(\030)1434 615 y Fi(1)1454 608 y Fo(.)42 b(It)23 b(is)f(easy)i(to)164 668 y(v)o(erify)e(that)i Fm(P)7 b Fo(\()p Fm(\030)498 675 y Fi(1)518 668 y Fo(\))26 b Fj(2)g Fm(L)p Fo(\()p Fm(L)707 650 y Fl(r)726 668 y Fo(\()p Ff(R)784 675 y Fl(y)802 668 y Fo(\)\),)f(the)e(space)h(of)f(b)q (ounded)i(linear)d(op)q(erators)j(in)164 729 y Fm(L)197 711 y Fl(r)216 729 y Fo(\()p Ff(R)274 736 y Fl(y)292 729 y Fo(\).)48 b(Moreo)o(v)o(er,)26 b(due)f(to)h(the)f(Lizorkin)g (theorem,)g Fm(p)p Fo(\()p Fm(\030)r Fo(\))i(is)e(a)h(m)o(ultipli)o(er) c(in)164 789 y Fm(L)197 771 y Fl(r)216 789 y Fo(\()p Ff(R)274 771 y Fi(2)291 789 y Fo(\).)37 b(Hence,)21 b(so)h(is)f(for)g Fm(P)7 b Fo(\()p Fm(\030)802 796 y Fi(1)823 789 y Fo(\))21 b(in)g(the)g(space)h Fm(L)1183 771 y Fl(r)1202 789 y Fo(\()p Ff(R)1260 796 y Fl(x)1279 789 y Fm(;)8 b(L)1334 771 y Fl(r)1353 789 y Fo(\()p Ff(R)1411 796 y Fl(y)1429 789 y Fo(\)\))22 b(=)h Fm(L)1583 771 y Fl(r)1602 789 y Fo(\()p Ff(R)1660 771 y Fi(2)1677 789 y Fo(\).)36 b(It)164 849 y(is)25 b(not)g(di\016cult)f(to)h(v)o(erify)e(that)j Fm(P)7 b Fo(\()p Fm(\030)910 856 y Fi(1)930 849 y Fo(\))25 b(dep)q(ends)g(con)o(tin)o(uously)g(on)g Fm(\030)1561 856 y Fi(1)1606 849 y Fo(with)g(re-)164 909 y(sp)q(ect)c(to)g(the)g (norm)f(in)h Fm(L)p Fo(\()p Fm(L)727 891 y Fl(r)746 909 y Fo(\()p Ff(R)804 916 y Fl(y)822 909 y Fo(\)\))f(at)i(an)o(y)f(p)q (oin)o(t)g Fm(\030)1195 916 y Fi(1)1236 909 y Fj(6)p Fo(=)h(0.)36 b(Therefore,)21 b(b)o(y)g(The-)164 969 y(orem)d(3.8)i(of)g (Ch.)g(7,)g([21],)g(w)o(e)f(see)g(that)h Fm(P)7 b Fo(\()p Fm(\030)1038 976 y Fi(1)1058 969 y Fo(\))20 b(is)f(also)h(a)g(m)o (ultiplier)c(in)j(the)h(space)164 1030 y Fm(L)197 1012 y Fl(r)216 1030 y Fo(\(\()p Fj(\000)p Fm(k)r(=)p Fo(2)p Fm(;)8 b(k)r(=)p Fo(2\))p Fm(;)g(L)539 1012 y Fl(r)559 1030 y Fo(\()p Ff(R)617 1037 y Fl(y)635 1030 y Fo(\)\))673 1037 y Fi(0)706 1030 y Fo(=)14 b Fm(L)791 1012 y Fl(r)810 1030 y Fo(\()p Fm(Q)868 1037 y Fl(k)889 1030 y Fo(\))908 1037 y Fi(0)944 1030 y Fo(considered)i(as)h(the)e(space)i(of)f Fm(k)r Fo(-p)q(erio)q(dic)g(in)164 1090 y Fm(x)g Fo(functions.)21 b(The)c(subscript)f(0)h(means)e(that)i(for)f(functions)h(from)e(this)h (space)g Fj(F)1704 1097 y Fl(k)q(;x)1755 1090 y Fm(u)164 1150 y Fo(v)m(anishes)h(at)g Fm(\030)437 1157 y Fi(1)472 1150 y Fo(=)d(0.)24 b(Since)15 b Fm(p)p Fo(\(0)p Fm(;)8 b(\030)823 1157 y Fi(2)844 1150 y Fo(\))15 b(=)f(0,)j(the)g(corresp)q (onding)h(m)o(ultipli)o(er)c(v)m(anishes)164 1210 y(on)20 b Fj(f)p Fm(u)e Fj(2)h Fm(L)391 1192 y Fl(r)411 1210 y Fo(\()p Fm(Q)469 1217 y Fl(k)490 1210 y Fo(\))f(:)h Fj(F)596 1217 y Fl(k)q(;x)647 1210 y Fm(u)f Fo(=)h(0)49 b(if)g Fm(\030)922 1217 y Fi(1)961 1210 y Fj(6)p Fo(=)18 b(0)p Fj(g)i Fo(and,)g(hence,)f(is)g(a)h(b)q(ounded)g(op)q(era-)164 1270 y(tor)f(on)h(the)f(en)o(tier)e(space)j Fm(L)709 1252 y Fl(r)728 1270 y Fo(\()p Fm(Q)786 1277 y Fl(k)807 1270 y Fo(\).)29 b(In)19 b(fact,)g(w)o(e)g(need)g(here)f(an)i (extension)e(of)h(that)164 1331 y(theorem)e(for)h(op)q(erator)h(v)m (alued)f(m)o(ultipliers)d(whic)o(h)j(ma)o(y)e(b)q(e)j(discon)o(tin)o (uous)f(at)g(the)164 1391 y(p)q(oin)o(t)e(0.)21 b(Ho)o(w)o(ev)o(er,)14 b(in)h(this)g(case)h(the)f(pro)q(of)i(presen)o(ted)e(in)g([21])g(w)o (orks)h(without)g(an)o(y)164 1451 y(c)o(hange.)237 1511 y(T)l(o)k(complete)c(the)j(pro)q(of)h(in)f(the)g(case)g(5)g Fm(<)f(p)h(<)g Fo(6)g(w)o(e)g(can)g(no)o(w)g(use)g(the)g(same)164 1571 y(reiteration)g(argumen)o(t,)h(as)g(in)g([6].)32 b(In)20 b(the)g(case)g(2)g Fm(<)g(p)h Fj(\024)f Fo(5)h(again)g(one)f (needs)g(to)164 1632 y(in)o(v)o(ok)o(e)12 b(the)h(argumen)o(ts)f(of)i (the)f(pro)q(of)i(of)e(Theorem)g(4.1,)g([5].)20 b(Here)13 b(w)o(e)g(ha)o(v)o(e)f(to)i(apply)164 1692 y(the)i(remark)f(on)i(op)q (erator)h(v)m(alued)f(m)o(ultipli)o(ers)d(to)j Fm(p)1161 1699 y Fi(1)1181 1692 y Fo(\()p Fm(\030)r Fo(\))e(=)g Fm(\030)1333 1674 y Fi(2)1331 1704 y(1)1353 1692 y Fm(p)p Fo(\()p Fm(\030)r Fo(\),)i Fm(p)1493 1699 y Fi(2)1514 1692 y Fo(\()p Fm(\030)r Fo(\))e(=)f Fm(\030)1663 1699 y Fi(2)1683 1692 y Fm(p)p Fo(\()p Fm(\030)r Fo(\),)164 1752 y(as)j(w)o(ell)e(as)i(to)f Fm(p)p Fo(\()p Fm(\030)r Fo(\))i(itself.)i Fb(2)237 1812 y Fo(W)l(e)c(need)g(also)h(the)f(follo) o(wing)g(additional)g(assumption:)164 1893 y Fp(\(N1\))24 b Fm(f)19 b Fj(2)14 b Fm(C)433 1875 y Fi(1)452 1893 y Fo(\()p Ff(R)p Fo(\))h Fe(and,)i(for)g(any)g Fm(v)f Fj(2)e Fm(L)944 1875 y Fi(2)964 1893 y Fo(\()p Ff(R)1021 1875 y Fi(2)1038 1893 y Fo(\))k Fe(such)f(that)1285 1853 y Fk(R)1309 1911 y Fd(R)1336 1901 y Fc(2)1360 1893 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)15 b(>)f Fo(0)p Fe(,)k(we)g(have)789 1941 y Fk(Z)816 2054 y Fd(R)843 2044 y Fc(2)868 2009 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)15 b(<)1052 1941 y Fk(Z)1079 2054 y Fd(R)1106 2044 y Fc(2)1131 2009 y Fm(f)1160 1988 y Fh(0)1172 2009 y Fo(\()p Fm(v)r Fo(\))p Fm(v)1262 1988 y Fi(2)286 2124 y Fe(and)381 2084 y Fk(R)404 2141 y Fd(R)431 2132 y Fc(2)456 2124 y Fm(f)5 b Fo(\()p Fm(tv)r Fo(\))p Fm(v)14 b(>)g Fo(0)50 b Fj(8)p Fm(t)12 b(>)i Fo(0)p Fe(.)164 2205 y Fo(Calculating)i(the)g(deriv)m(ativ)o(e)f (of)h Fm(t)805 2187 y Fh(\000)p Fi(1)860 2165 y Fk(R)884 2223 y Fd(R)911 2213 y Fc(2)935 2205 y Fm(f)5 b Fo(\()p Fm(tv)r Fo(\))p Fm(v)r Fo(,)15 b(w)o(e)h(see)g(that)g(\(N1\))g(implies) e(\(N\).)237 2265 y(Let)j(us)f(in)o(tro)q(duce)g(the)g(functional)556 2363 y Fm(L)589 2370 y Fl(k)611 2363 y Fo(\()p Fm(v)r Fo(\))d(=)740 2296 y Fk(Z)767 2408 y Fl(Q)795 2414 y Fg(k)816 2363 y Fo([)835 2330 y(1)p 835 2352 25 2 v 835 2398 a(2)864 2363 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)12 b Fj(\000)f Fm(F)c Fo(\()p Fm(v)r Fo(\)])p Fm(;)55 b(v)15 b Fj(2)f Fm(X)1355 2370 y Fl(k)1377 2363 y Fm(:)164 2482 y Fo(As)i(w)o(e)g(ha)o(v)o(e)f(seen,)h Fm(L)573 2489 y Fl(k)608 2482 y Fo(=)e Fm(J)687 2489 y Fl(k)724 2482 y Fo(on)j Fm(S)822 2489 y Fl(k)860 2482 y Fo(and)g Fm(L)988 2489 y Fl(k)1009 2482 y Fo(\()p Fm(v)r Fo(\))c Fj(\025)h Fo(0,)i Fj(8)p Fm(v)e Fj(2)g Fm(X)1347 2489 y Fl(k)1369 2482 y Fo(.)949 2607 y(10)p eop %%Page: 11 12 11 11 bop 164 315 a Fp(Lemma)16 b(4)24 b Fe(Under)18 b(Assumption)h Fo(\(N1\))p Fe(,)f Fm(L)998 322 y Fl(k)1019 315 y Fo(\()p Fm(tv)r Fo(\))f Fe(is)h(a)g(strictly)g(incr)n(e)n(asing)h (function)164 376 y(of)e Fm(t)d(>)f Fo(0)p Fe(,)18 b(pr)n(ovide)n(d)554 336 y Fk(R)577 393 y Fl(Q)605 399 y Fg(k)634 376 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)15 b(>)f Fo(0)p Fe(.)237 490 y(Pr)n(o)n(of)9 b Fo(.)21 b(It)16 b(follo)o(ws)g(immedi)o(ately)d (from)i(the)h(follo)o(wing)g(elemen)o(tary)d(iden)o(tit)o(y)462 588 y Fm(d)p 454 610 43 2 v 454 656 a(dt)501 622 y(L)534 629 y Fl(k)556 622 y Fo(\()p Fm(tv)r Fo(\))g(=)716 588 y(1)p 707 610 V 707 656 a(2)p Fm(t)754 622 y Fo([)768 554 y Fk(Z)796 667 y Fl(Q)824 673 y Fg(k)853 622 y Fm(f)882 601 y Fh(0)894 622 y Fo(\()p Fm(tv)r Fo(\))p Fm(t)994 601 y Fi(2)1012 622 y Fm(v)g Fj(\000)1098 554 y Fk(Z)1126 667 y Fl(Q)1154 673 y Fg(k)1183 622 y Fm(f)5 b Fo(\()p Fm(tv)r Fo(\))p Fm(tv)r Fo(])p Fm(:)95 b Fb(2)237 761 y Fo(W)l(e)11 b(need)g(also)h(the)f(follo)o(wing)g(dual)g(c)o (haracterization)g(of)g(ground)i(tra)o(v)o(eling)d(w)o(a)o(v)o(es)164 876 y Fp(Lemma)16 b(5)24 b Fe(Supp)n(ose)14 b(Assumptions)h Fo(1)906 858 y Fi(0)926 876 y Fe({)p Fo(4)975 858 y Fi(0)1009 876 y Fe(and)g Fo(\(N1\))f Fe(to)g(b)n(e)h(satis\014e)n(d.)21 b(F)l(or)14 b(nonzer)n(o)164 936 y Fm(u)g Fj(2)g Fm(X)293 943 y Fl(k)314 936 y Fe(,)k Fm(k)e Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(])p Fe(,)17 b(the)h(fol)r(lowing)h(statements)g(ar)n(e)e(e) n(quivalent:)164 1038 y Fo(\()p Fm(i)p Fo(\))24 b Fm(u)17 b Fe(is)g(a)h(gr)n(ound)f(wave,)164 1139 y Fo(\()p Fm(ii)p Fo(\))24 b Fm(I)282 1146 y Fl(k)302 1139 y Fo(\()p Fm(u)p Fo(\))14 b(=)g(0)k Fe(and)f Fm(L)603 1146 y Fl(k)625 1139 y Fo(\()p Fm(u)p Fo(\))c(=)h Fm(m)799 1146 y Fl(k)834 1139 y Fo(=)g(inf)s Fj(f)p Fm(L)1003 1146 y Fl(k)1024 1139 y Fo(\()p Fm(v)r Fo(\))f(:)h Fm(v)h Fj(2)f Fm(S)1245 1146 y Fl(k)1266 1139 y Fj(g)p Fe(,)164 1241 y Fo(\()p Fm(iii)p Fo(\))23 b Fm(I)298 1248 y Fl(k)319 1241 y Fo(\()p Fm(u)p Fo(\))14 b(=)f(0)i(=)e(sup)q Fj(f)p Fm(I)661 1248 y Fl(k)682 1241 y Fo(\()p Fm(v)r Fo(\))g(:)g Fm(v)i Fj(2)f Fm(X)912 1248 y Fl(k)934 1241 y Fm(;)8 b(L)989 1248 y Fl(k)1010 1241 y Fo(\()p Fm(v)r Fo(\))14 b(=)f Fm(m)1182 1248 y Fl(k)1203 1241 y Fj(g)p Fe(.)237 1355 y(Pr)n(o)n(of)c Fo(.)21 b(Implication)14 b(\()p Fm(i)p Fo(\))f Fj(\))h Fo(\()p Fm(ii)p Fo(\))i(is)g(pro)o(v)o(ed)f(in)h(Section)g(2.)237 1415 y(T)l(o)i(pro)o(v)o(e)f(\()p Fm(ii)p Fo(\))f Fj(\))g Fo(\()p Fm(i)p Fo(\))h(assume)g(that)h Fm(u)e Fj(2)h Fm(X)1080 1422 y Fl(k)1120 1415 y Fo(satis\014es)h(\()p Fm(ii)p Fo(\).)24 b(Since)17 b Fm(J)1567 1422 y Fl(k)1605 1415 y Fo(=)f Fm(L)1692 1422 y Fl(k)1731 1415 y Fo(on)164 1475 y Fm(S)194 1482 y Fl(k)215 1475 y Fo(,)g(there)g(exists)f(a)i (Lagrange)h(m)o(ultiplie)o(r)c Fm(\025)i Fo(suc)o(h)g(that)808 1585 y Fm(\025I)862 1565 y Fh(0)858 1598 y Fl(k)880 1585 y Fo(\()p Fm(u)p Fo(\))d(=)h Fm(J)1043 1565 y Fh(0)1038 1598 y Fl(k)1059 1585 y Fo(\()p Fm(u)p Fo(\))p Fm(:)164 1695 y Fo(Then)588 1756 y Fm(\025)p Fj(h)p Fm(I)661 1735 y Fh(0)657 1768 y Fl(k)679 1756 y Fo(\()p Fm(u)p Fo(\))p Fm(;)8 b(u)p Fj(i)13 b Fo(=)h Fj(h)p Fm(J)930 1735 y Fh(0)925 1768 y Fl(k)947 1756 y Fo(\()p Fm(u)p Fo(\))p Fm(;)8 b(u)p Fj(i)13 b Fo(=)h Fm(I)1169 1763 y Fl(k)1190 1756 y Fo(\()p Fm(u)p Fo(\))f(=)h(0)p Fm(:)164 1843 y Fo(On)i(the)g(other)h(hand)470 1965 y Fj(h)p Fm(I)515 1944 y Fh(0)511 1977 y Fl(k)533 1965 y Fo(\()p Fm(u)p Fo(\))p Fm(;)8 b(u)p Fj(i)13 b Fo(=)h(2)p Fj(k)p Fm(u)p Fj(k)835 1944 y Fi(2)835 1977 y Fl(k)867 1965 y Fj(\000)917 1897 y Fk(Z)945 2010 y Fl(Q)973 2016 y Fg(k)1002 1965 y Fm(f)1031 1944 y Fh(0)1043 1965 y Fo(\()p Fm(u)p Fo(\))p Fm(u)1137 1944 y Fi(2)1167 1965 y Fj(\000)1217 1897 y Fk(Z)1245 2010 y Fl(Q)1273 2016 y Fg(k)1302 1965 y Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)14 b Fo(=)571 2138 y(=)g(2)p Fm(I)669 2145 y Fl(k)690 2138 y Fo(\()p Fm(u)p Fo(\))d(+)816 2070 y Fk(Z)844 2183 y Fl(Q)872 2189 y Fg(k)901 2138 y Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)11 b Fj(\000)1085 2070 y Fk(Z)1112 2183 y Fl(Q)1140 2189 y Fg(k)1170 2138 y Fm(f)1199 2117 y Fh(0)1211 2138 y Fo(\()p Fm(u)p Fo(\))p Fm(u)1305 2117 y Fi(2)1338 2138 y Fo(=)687 2288 y(=)739 2220 y Fk(Z)766 2333 y Fl(Q)794 2339 y Fg(k)824 2288 y Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)10 b Fj(\000)1007 2220 y Fk(Z)1035 2333 y Fl(Q)1063 2339 y Fg(k)1092 2288 y Fm(f)1121 2268 y Fh(0)1133 2288 y Fo(\()p Fm(u)p Fo(\))p Fm(u)1227 2268 y Fi(2)1246 2288 y Fm(:)164 2411 y Fo(Ho)o(w)o(ev)o(er,) 377 2371 y Fk(R)400 2428 y Fl(Q)428 2434 y Fg(k)457 2411 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)19 b(>)e Fo(0)i(on)g Fm(S)791 2418 y Fl(k)831 2411 y Fo(and,)g(due)f(to)h(\(N1\),)f Fj(h)p Fm(I)1274 2393 y Fh(0)1270 2424 y Fl(k)1291 2411 y Fo(\()p Fm(u)p Fo(\))p Fm(;)8 b(u)p Fj(i)17 b Fm(<)h Fo(0.)28 b(Therefore,)164 2471 y Fm(\025)14 b Fo(=)g(0)j(and)g Fm(u)f Fo(is)g(a)g(ground)i(w)o(a)o(v)o(e.)949 2607 y(11)p eop %%Page: 12 13 12 12 bop 237 315 a Fo(No)o(w)18 b(let)g(us)g(pro)o(v)o(e)g(\()p Fm(ii)p Fo(\))e Fj(\))h Fo(\()p Fm(iii)p Fo(\).)26 b(F)l(or)18 b Fm(u)g Fo(as)h(in)f(\()p Fm(ii)p Fo(\),)f Fm(I)1286 322 y Fl(k)1307 315 y Fo(\()p Fm(u)p Fo(\))g(=)g(0.)28 b(Assume)16 b(that)164 376 y(there)h(is)h Fm(v)g Fj(2)e Fm(X)472 383 y Fl(k)512 376 y Fo(suc)o(h)i(that)g Fm(L)764 383 y Fl(k)785 376 y Fo(\()p Fm(v)r Fo(\))e(=)h Fm(m)963 383 y Fl(k)1001 376 y Fo(and)i Fm(I)1120 383 y Fl(k)1141 376 y Fo(\()p Fm(v)r Fo(\))c Fm(<)i Fo(0.)26 b(Then)1468 336 y Fk(R)1510 376 y Fm(Q)1549 383 y Fl(k)1570 376 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)17 b(>)g Fo(0)164 436 y(and)i(there)f(exists)g Fm(t)543 443 y Fi(0)580 436 y Fj(2)f Fo(\(0)p Fm(;)8 b Fo(1\))19 b(suc)o(h)g(that)g Fm(I)1000 443 y Fl(k)1020 436 y Fo(\()p Fm(t)1057 443 y Fi(0)1077 436 y Fm(v)r Fo(\))e(=)g(0.)28 b(By)18 b(Lemma)e(4,)j Fm(L)1605 443 y Fl(k)1627 436 y Fo(\()p Fm(t)1664 443 y Fi(0)1683 436 y Fm(v)r Fo(\))e Fm(<)164 496 y(L)197 503 y Fl(k)218 496 y Fo(\()p Fm(v)r Fo(\))d(=)f Fm(m)390 503 y Fl(k)411 496 y Fo(,)j(whic)o(h)g(is)g(imp)q(ossible.)237 556 y(Finally)l(,)g(w)o(e)h(pro)o(v)o(e)f(\()p Fm(iii)p Fo(\))f Fj(\))h Fo(\()p Fm(ii)p Fo(\).)23 b(Let)18 b Fm(u)d Fj(2)h Fm(X)1119 563 y Fl(k)1159 556 y Fo(satis\014es)h(\()p Fm(iii)p Fo(\).)24 b(Then,)17 b Fm(L)1641 563 y Fl(k)1663 556 y Fo(\()p Fm(u)p Fo(\))e Fj(\025)164 616 y Fm(m)207 623 y Fl(k)228 616 y Fo(.)22 b(Assume)15 b(that)i Fm(L)584 623 y Fl(k)605 616 y Fo(\()p Fm(u)p Fo(\))d Fm(>)g(m)780 623 y Fl(k)801 616 y Fo(.)22 b(Again)16 b(w)o(e)g(ha)o(v)o(e)1164 576 y Fk(R)1187 634 y Fl(Q)1215 640 y Fg(k)1245 616 y Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)14 b(>)g Fo(0.)22 b(By)16 b(Lemma)e(4,)164 682 y(there)k(exists)h Fm(t)447 689 y Fi(0)485 682 y Fj(2)g Fo(\(0)p Fm(;)8 b Fo(1\))19 b(suc)o(h)g(that)h Fm(L)919 689 y Fl(k)940 682 y Fo(\()p Fm(t)977 689 y Fi(0)997 682 y Fm(u)p Fo(\))e(=)h Fm(m)1162 689 y Fl(k)1183 682 y Fo(.)29 b(Ho)o(w)o(ev)o(er,)18 b Fm(I)1462 689 y Fl(k)1483 682 y Fo(\()p Fm(t)1520 689 y Fi(0)1539 682 y Fm(u)p Fo(\))g Fm(>)h Fo(0)g(and)164 742 y(this)d(con)o(tradicts)g(\()p Fm(iii)p Fo(\).)k Fb(2)p Fo(.)237 802 y(No)o(w)d(w)o(e)g(are)g(ready)g(to)g(pro)o(v)o(e)f (the)h(symetry)e(prop)q(ert)o(y)i(for)g(all)g(kinds)g(of)g(ground)164 862 y(w)o(a)o(v)o(es)e(w)o(e)g(consider.)21 b(As)15 b(in)g([6],)g(w)o (e)g(use)h(the)f(approac)o(h)h(suggested)h(in)e([12])g(\(see)g(also)164 922 y([22]\).)164 1031 y Fp(Theorem)i(3)24 b Fe(In)19 b(addition)g(to)g(Assumptions)h Fo(1)1085 1013 y Fi(0)1105 1031 y Fe({)p Fo(4)1154 1013 y Fi(0)1193 1031 y Fe(and)f Fo(\(N1\))p Fe(,)h(supp)n(ose)e(that)h Fm(f)k Fj(2)164 1091 y Fm(C)203 1073 y Fi(2)222 1091 y Fo(\()p Ff(R)p Fo(\))p Fe(.)30 b(In)22 b(the)f(c)n(ase)h Fo(2)f Fm(<)f(p)h Fj(\024)g Fo(5)g Fe(assume)g(also)h(that)f(ine)n(quality)h(\(12\))f(is) g(ful\014le)n(d.)164 1151 y(Then)15 b(any)f(gr)n(ound)g(wave)h Fm(u)f Fj(2)g Fm(X)781 1158 y Fl(k)803 1151 y Fe(,)h Fm(k)h Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(])p Fe(,)14 b(is)g(symmetric)g(with)h(r)n(esp)n(e)n(ct)e(to)i(some)164 1212 y(line)k Fo(\001)13 b(=)h Fj(f)p Fo(\()p Fm(x;)8 b(y)r Fo(\))13 b Fj(2)h Ff(R)603 1193 y Fi(2)633 1212 y Fo(:)g Fm(y)h Fo(=)f Fm(b)p Fj(g)p Fe(.)237 1320 y(Pr)n(o)n(of)9 b Fo(.)21 b(Cho)q(ose)d Fm(b)e Fo(in)g(suc)o(h)g(a)g(w)o(a)o(y)g(that) 367 1379 y Fk(Z)395 1492 y Fi(\001)424 1482 y Fc(+)449 1492 y Fh(\\)p Fl(Q)501 1498 y Fg(k)522 1447 y Fo([)541 1413 y(1)p 541 1435 25 2 v 541 1481 a(2)570 1447 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)12 b Fj(\000)f Fm(F)c Fo(\()p Fm(v)r Fo(\)])12 b(=)929 1379 y Fk(Z)957 1492 y Fi(\001)986 1482 y Fa(\000)1012 1492 y Fh(\\)p Fl(Q)1064 1498 y Fg(k)1085 1447 y Fo([)1104 1413 y(1)p 1104 1435 V 1104 1481 a(2)1133 1447 y Fm(f)5 b Fo(\()p Fm(v)r Fo(\))p Fm(v)12 b Fj(\000)f Fm(F)c Fo(\()p Fm(v)r Fo(\)])12 b(=)1497 1413 y Fm(m)1540 1420 y Fl(k)p 1497 1435 64 2 v 1517 1481 a Fo(2)1566 1447 y Fm(;)164 1585 y Fo(where)i(\001)344 1567 y Fi(+)388 1585 y Fo(and)g(\001)521 1567 y Fh(\000)565 1585 y Fo(are)g(corresp)q (onding)i(upp)q(er)e(and)h(lo)o(w)o(er)f(half-planes.)21 b(Let)14 b Fm(u)1689 1567 y Fh(\006)1733 1585 y Fo(b)q(e)164 1645 y(a)20 b(symme)o(tric)c(\(with)j(resp)q(ect)g(to)g(\001\))g (function)g(suc)o(h)g(that)g Fm(u)1335 1627 y Fh(\006)1383 1645 y Fo(=)g Fm(u)g Fo(on)h(\001)1599 1627 y Fh(\006)1628 1645 y Fo(.)30 b(Then)164 1706 y Fm(u)192 1688 y Fh(\006)235 1706 y Fj(2)14 b Fm(X)322 1713 y Fl(k)360 1706 y Fo(and)734 1766 y Fm(L)767 1773 y Fl(k)789 1766 y Fo(\()p Fm(u)836 1745 y Fh(\006)865 1766 y Fo(\))g(=)f Fm(L)982 1773 y Fl(k)1004 1766 y Fo(\()p Fm(u)p Fo(\))g(=)h Fm(m)1178 1773 y Fl(k)1199 1766 y Fm(:)164 1851 y Fo(By)i(Lemma)e(5,)i Fm(I)491 1858 y Fl(k)512 1851 y Fo(\()p Fm(u)559 1833 y Fh(\006)588 1851 y Fo(\))e Fj(\024)f Fo(0.)22 b(On)16 b(the)g(other)h(hand,)654 1956 y Fm(I)676 1963 y Fl(k)697 1956 y Fo(\()p Fm(u)744 1935 y Fi(+)773 1956 y Fo(\))11 b(+)g Fm(I)874 1963 y Fl(k)895 1956 y Fo(\()p Fm(u)942 1935 y Fh(\000)972 1956 y Fo(\))i(=)h(2)p Fm(I)1102 1963 y Fl(k)1123 1956 y Fo(\()p Fm(u)p Fo(\))g(=)g(0)p Fm(:)164 2061 y Fo(Using)i(Lemma)e(5,)i(w)o(e)g(conclude)g(that)g Fm(u)935 2043 y Fh(\006)981 2061 y Fo(is)g(a)h(ground)g(w)o(a)o(v)o(e.) 237 2121 y(T)l(o)e(conclude)f(that)h Fm(u)638 2103 y Fh(\006)681 2121 y Fo(=)f Fm(u)g Fo(and,)h(hence,)f(complete)e(the)i (pro)q(of)i(it)e(is)g(su\016cien)o(t)g(to)164 2181 y(use)i(the)f(same)f (unique)h(con)o(tin)o(uation)g(result,)g(as)h(in)f([6],)g(and)h(just)g (here)f(w)o(e)g(need)g(the)164 2242 y(assumption)h Fm(f)j Fj(2)14 b Fm(C)551 2224 y Fi(2)570 2242 y Fo(\()p Ff(R)p Fo(\))f(and)k(Lemma)c(3.)22 b(Remark)14 b(that)j(a)f(p)q(erio)q(dic)g (v)o(ersion)f(\(with)164 2302 y(\005)e(=)h(\001)307 2284 y Fh(\006)336 2302 y Fo(\))i(of)h(unique)e(con)o(tin)o(uation)h (Theorem)e(A.1,)i([6)o(],)g(can)g(b)q(e)g(pro)o(v)o(ed)f(exactly)g(as) 164 2362 y(that)i(theorem)d(itself.)20 b Fb(2)237 2422 y Fo(In)c(addition,)g(w)o(e)f(form)o(ulate)g(the)g(follo)o(wing)h (direct)f(generalization)g(of)i(results)f(of)164 2482 y([6])g(for)g(deca)o(y)g(of)g(solitary)g(w)o(a)o(v)o(es.)949 2607 y(12)p eop %%Page: 13 14 13 13 bop 164 315 a Fp(Theorem)17 b(4)24 b Fe(Supp)n(ose)e(Assumptions) g Fo(1)960 297 y Fi(0)1002 315 y Fe(and)g Fo(2)1125 297 y Fi(0)1167 315 y Fe(to)g(b)n(e)g(satis\014e)n(d.)36 b(L)n(et)21 b Fm(u)h Fj(2)g Fm(X)1746 322 y Fl(k)1768 315 y Fe(,)164 376 y Fm(k)16 b Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(])p Fe(,)17 b(b)n(e)h(a)f(tr)n(aveling)i(wave.)k(If)17 b Fm(k)f Fo(=)e Fj(1)p Fe(,)j(then)627 486 y Fm(r)650 465 y Fi(2)670 486 y Fm(u)d Fj(2)g Fm(L)792 465 y Fh(1)829 486 y Fo(\()p Ff(R)887 465 y Fi(2)904 486 y Fo(\))p Fm(;)108 b(r)1068 465 y Fi(2)1101 486 y Fo(=)14 b Fm(x)1181 465 y Fi(2)1212 486 y Fo(+)d Fm(y)1287 465 y Fi(2)1306 486 y Fm(:)164 596 y Fe(If)17 b Fo(0)d Fm(<)g(k)i(<)e Fj(1)p Fe(,)j(then)815 656 y Fm(y)841 635 y Fi(2)861 656 y Fm(u)c Fj(2)h Fm(L)982 635 y Fh(1)1020 656 y Fo(\()p Fm(Q)1078 663 y Fl(k)1099 656 y Fo(\))p Fm(:)237 770 y Fo(The)j(pro)q(of)h(is)f (essen)o(tially)e(the)i(same)f(as)h(in)g([6].)22 b(In)17 b(the)g(case)f Fm(k)h(<)e Fj(1)i Fo(one)g(needs)164 830 y(only)k(to)h(use)f(the)g(partially)g(p)q(erio)q(dic)g(F)l(ourier)g (transform,)h(as)g(in)f(Lemma)e(3.)37 b(Let)164 890 y(us)19 b(remark)e(that)j(the)e(classical)h(rational)g(KP-solitons)g(deca)o(y)f (exactly)g(as)i Fm(r)1628 897 y Fh(\000)p Fi(2)1675 890 y Fo(.)29 b(On)164 951 y(the)18 b(other)h(hand,)h(in)e([24])g(it)h(is)f (constructed)h(a)g(family)d(of)j(tra)o(v)o(eling)e(w)o(a)o(v)o(es)h (for)h(the)164 1011 y(original)c(KP-I)g(equations)g(whic)o(h)f(are)h(p) q(erio)q(dic)g(in)f Fm(x)h Fo(and)h(deca)o(y)e(exp)q(onen)o(tially)f (fast)164 1071 y(with)i(resp)q(ect)f(to)h(the)g(transv)o(erse)f(v)m (ariable.)21 b(Th)o(us,)15 b(it)f(seems)g(that)h(the)f(statemen)o(t)g (of)164 1131 y(Theorem)h(4)i(is)f(not)h(exact)f(in)h(the)f(case)g Fm(k)h(<)d Fj(1)p Fo(,)i(while)f(it)i(is)f(so)h(in)f(the)g(case)h Fm(k)f Fo(=)f Fj(1)p Fo(.)164 1191 y(It)h(is)g(also)h(unkno)o(wn)f(are) h(Zaitsev's)e(solutions)i(ground)g(w)o(a)o(v)o(es,)e(or)i(not.)164 1358 y Fn(4)81 b(On)26 b(Nonexistence)g(of)h(T)-7 b(ra)n(v)n(eling)25 b(W)-7 b(a)n(v)n(es)164 1467 y Fo(In)13 b(this)g(section)f(w)o(e)h (turn)g(to)h(general)e(KP)i(equations)f(\(3\),)g(with)g Fm(")h Fo(=)g Fj(\006)p Fo(1,)f(and)h(discuss)164 1527 y(the)i(nonexistence)e(problem.)20 b(W)l(e)15 b(use)h(the)g(same)f (approac)o(h)h(as)h(in)e([5].Ho)o(w)o(ev)o(er,)e(the)164 1588 y(case)i(of)g(p)q(erio)q(dic)g(w)o(a)o(v)o(es)g(is)f(more)g(in)o (v)o(olv)o(ed)f(\(see)i(the)g(pro)q(of)h(of)f(Lemma)e(6\).)21 b(Here)14 b(w)o(e)164 1648 y(consider)i(tra)o(v)o(eling)f(w)o(a)o(v)o (es)g(b)q(elonging)i(to)g(the)f(space)250 1758 y Fm(Y)278 1765 y Fl(k)313 1758 y Fo(=)e Fj(f)p Fm(u)g Fj(2)g Fm(X)519 1765 y Fl(k)554 1758 y Fo(:)g Fm(u)f Fj(2)h Fm(H)714 1737 y Fi(1)734 1758 y Fo(\()p Fm(Q)792 1765 y Fl(k)813 1758 y Fo(\))p Fm(;)8 b(u)882 1765 y Fl(xx)924 1758 y Fm(;)g(D)987 1737 y Fh(\000)p Fi(1)986 1772 y Fl(x;k)1037 1758 y Fm(u)1065 1765 y Fl(y)q(y)1118 1758 y Fj(2)14 b Fm(L)1198 1737 y Fi(2)1218 1758 y Fo(\()p Fm(Q)1276 1765 y Fl(k)1297 1758 y Fo(\))p Fm(;)8 b(f)d Fo(\()p Fm(u)p Fo(\))p Fm(u)13 b Fj(2)h Fm(L)1554 1737 y Fi(1)1574 1758 y Fo(\()p Fm(Q)1632 1765 y Fl(k)1653 1758 y Fo(\))p Fj(g)164 1868 y Fo(if)i Fm(k)g(<)d Fj(1)p Fo(,)j(and)185 1978 y Fm(Y)25 b Fo(=)14 b Fm(Y)318 1985 y Fh(1)369 1978 y Fo(=)g Fj(f)p Fm(u)g Fj(2)g Fm(X)k Fo(:)13 b Fm(u)h Fj(2)g Fm(H)753 1957 y Fi(1)773 1978 y Fo(\()p Ff(R)831 1957 y Fi(2)847 1978 y Fo(\))p Fm(;)8 b(u)916 1985 y Fl(xx)958 1978 y Fm(;)g(D)1021 1957 y Fh(\000)p Fi(1)1020 1992 y Fl(x;k)1071 1978 y Fm(u)1099 1985 y Fl(y)q(y)1152 1978 y Fj(2)14 b Fm(L)1232 1957 y Fi(2)1232 1990 y Fl(loc)1278 1978 y Fo(\()p Ff(R)1336 1957 y Fi(2)1352 1978 y Fo(\))p Fm(;)8 b(f)d Fo(\()p Fm(u)p Fo(\))p Fm(u)14 b Fj(2)g Fm(L)1610 1957 y Fi(1)1630 1978 y Fo(\()p Ff(R)1688 1957 y Fi(2)1704 1978 y Fo(\))p Fj(g)p Fm(:)237 2088 y Fo(First,)h(w)o(e)h (collect)f(some)g(useful)h(iden)o(tities.)164 2202 y Fp(Lemma)g(6)24 b Fe(Supp)n(ose)g(that)f Fm(f)29 b Fe(satis\014es)c (Assumptions)e Fo(1)1272 2184 y Fi(0)1316 2202 y Fe(and)h Fo(2)1441 2184 y Fi(0)1461 2202 y Fe(.)41 b(L)n(et)23 b Fm(u)i Fj(2)g Fm(Y)1746 2209 y Fl(k)1768 2202 y Fe(,)164 2262 y Fm(k)16 b Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(])p Fe(,)17 b(b)n(e)h(a)f(solution)h(of)g(e)n(quations)g(\(3\).)k(Then)514 2334 y Fk(Z)542 2447 y Fl(Q)570 2453 y Fg(k)591 2402 y Fo([)612 2368 y Fm(c)p 610 2390 25 2 v 610 2436 a Fo(2)638 2402 y Fm(u)666 2381 y Fi(2)697 2402 y Fo(+)751 2368 y(3)p 751 2390 V 751 2436 a(2)780 2402 y Fm(u)808 2381 y Fi(2)808 2414 y Fl(x)841 2402 y Fo(+)11 b Fm(")918 2368 y(v)944 2350 y Fi(2)p 918 2390 46 2 v 928 2436 a Fo(2)979 2402 y Fj(\000)g Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))11 b(+)g Fm(F)c Fo(\()p Fm(u)p Fo(\)])12 b(=)i(0)p Fm(;)263 b Fo(\(15\))949 2607 y(13)p eop %%Page: 14 15 14 14 bop 606 271 a Fk(Z)633 384 y Fl(Q)661 390 y Fg(k)682 339 y Fo([)703 305 y Fm(c)p 701 327 25 2 v 701 373 a Fo(2)730 339 y Fm(u)758 318 y Fi(2)789 339 y Fo(+)843 305 y(1)p 843 327 V 843 373 a(2)872 339 y Fm(u)900 318 y Fi(2)900 351 y Fl(x)933 339 y Fo(+)11 b Fm(")1010 305 y(v)1036 287 y Fi(2)p 1010 327 46 2 v 1020 373 a Fo(2)1071 339 y(+)g Fm(F)c Fo(\()p Fm(u)p Fo(\)])12 b(=)i(0)p Fm(;)355 b Fo(\(16\))624 421 y Fk(Z)652 534 y Fl(Q)680 540 y Fg(k)701 489 y Fo([)p Fm(cu)764 469 y Fi(2)794 489 y Fo(+)11 b Fm(u)871 469 y Fi(2)871 502 y Fl(x)904 489 y Fj(\000)g Fm("v)1003 469 y Fi(2)1033 489 y Fj(\000)f Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)p Fo(])14 b(=)f(0)p Fm(:)374 b Fo(\(17\))237 636 y Fe(Pr)n(o)n(of)9 b Fo(.)23 b(First,)16 b(w)o(e)g(remark)f(that,)i (for)g(an)o(y)g Fm(k)r Fo(,)f(\(17\))i(is)e(an)i(exten)o(tion)d(to)i (the)g(case)164 696 y Fm(")f Fo(=)h Fj(\006)p Fo(1)h(of)g Fm(I)418 703 y Fl(k)439 696 y Fo(\()p Fm(u)p Fo(\))e(=)h(0)h(stated)g (in)g(Section)f(2.)27 b(Therefore,)17 b(w)o(e)h(concen)o(trate)f(at)h (\(15\))164 756 y(and)f(\(16\))g(only)l(.)237 817 y(In)22 b(the)h(case)g(of)f(solitary)h(w)o(a)o(v)o(es)f(\()p Fm(k)k Fo(=)f Fj(1)p Fo(\))e(the)f(calculations)g(carried)g(out)h(in) 164 877 y(the)d(pro)q(of)h(of)g(Theorem)e(1.1,)i([5],)f(w)o(ork)g (equally)f(w)o(ell)g(for)i(general)f(nonlinearities.)164 937 y(Therefore,)15 b(w)o(e)h(lo)q(ok)h(at)f(p)q(erio)q(dic)g(w)o(a)o (v)o(es)g(\()p Fm(k)g(<)e Fj(1)p Fo(\).)237 997 y(Fixed)19 b Fm(\024)h Fj(2)g Fo(\(0)p Fm(;)8 b Fo(1\),)21 b(let)e Fm(')727 1004 y Fl(T)774 997 y Fj(2)h Fm(C)866 979 y Fh(1)862 1010 y Fi(0)903 997 y Fo(\()p Ff(R)p Fo(\))d(b)q(e)j(a)g (nonnegativ)o(e)f(function)h(suc)o(h)f(that)164 1057 y Fm(')196 1064 y Fl(T)237 1057 y Fo(=)14 b(1)f(on)h([)p Fj(\000)p Fm(T)5 b(=)p Fo(2)p Fm(;)j(T)d(=)p Fo(2],)13 b Fm(')703 1064 y Fl(T)731 1057 y Fo(\()p Fm(x)p Fo(\))g(=)h(0)g(if)e Fj(j)p Fm(x)p Fj(j)h(\025)h Fo(\()p Fm(T)d Fo(+)t Fm(T)1200 1039 y Fl(\024)1222 1057 y Fo(\))p Fm(=)p Fo(2,)j(and)f Fm(')1440 1039 y Fi(\()p Fl(j)r Fi(\))1486 1057 y Fo(\()p Fm(x)p Fo(\))h Fj(\024)f Fm(C)1653 1064 y Fl(j)1671 1057 y Fm(=)p Fj(j)p Fm(x)p Fj(j)1751 1039 y Fl(j)1769 1057 y Fo(,)164 1118 y Fm(j)18 b Fo(=)d(1)p Fm(;)8 b Fo(2)p Fm(;)g(:)g(:)g(:)p Fo(,)17 b(if)f Fm(T)5 b(=)p Fo(2)16 b Fj(\024)f(j)p Fm(x)p Fj(j)g(\024)g Fo(\()p Fm(T)j Fo(+)11 b Fm(T)909 1100 y Fl(\024)931 1118 y Fo(\))p Fm(=)p Fo(2)18 b(\(the)e(construction)i(of)f(suc)o(h)g(a)g(function)164 1178 y(will)e(b)q(e)h(giv)o(en)g(later)g(on\).)237 1238 y(Multiplying)h(the)i(\014rst)g(equation)g(\(3\))h(b)o(y)e Fm(x')1109 1245 y Fl(T)1136 1238 y Fm(u)h Fo(and)h(in)o(tegrating)e(o)o (v)o(er)g Ff(R)1677 1220 y Fi(2)1694 1238 y Fo(,)h(w)o(e)164 1298 y(get,)d(after)g(a)h(n)o(um)o(b)q(er)d(of)j(in)o(tegrations)f(b)o (y)g(parts,)431 1396 y Fm(c)p 429 1418 25 2 v 429 1464 a Fo(2)467 1362 y Fk(Z)525 1430 y Fm(')557 1437 y Fl(T)584 1430 y Fm(u)612 1409 y Fi(2)643 1430 y Fj(\000)693 1362 y Fk(Z)751 1430 y Fm(')783 1437 y Fl(T)810 1430 y Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))11 b(+)993 1362 y Fk(Z)1051 1430 y Fm(')1083 1437 y Fl(T)1111 1430 y Fm(F)c Fo(\()p Fm(u)p Fo(\))j(+)1280 1396 y(3)p 1280 1418 V 1280 1464 a(2)1318 1362 y Fk(Z)1376 1430 y Fm(')1408 1437 y Fl(T)1435 1430 y Fm(u)1463 1409 y Fi(2)1463 1442 y Fl(x)1485 1430 y Fo(+)354 1594 y(+)p Fm(")420 1561 y Fo(1)p 420 1583 V 420 1628 a(2)458 1527 y Fk(Z)516 1594 y Fm(')548 1601 y Fl(T)575 1594 y Fm(v)601 1574 y Fi(2)632 1594 y Fo(+)686 1561 y(1)p 686 1583 V 686 1628 a(2)723 1527 y Fk(Z)781 1594 y Fm(x')841 1574 y Fh(0)841 1607 y Fl(T)869 1594 y Fm(u)897 1574 y Fi(2)927 1594 y Fj(\000)977 1527 y Fk(Z)1035 1594 y Fm(x')1095 1574 y Fh(0)1095 1607 y Fl(T)1122 1594 y Fm(uf)5 b Fo(\()p Fm(u)p Fo(\))11 b(+)1305 1527 y Fk(Z)1363 1594 y Fm(x')1423 1574 y Fh(0)1423 1607 y Fl(T)1450 1594 y Fm(F)c Fo(\()p Fm(u)p Fo(\)+)360 1736 y(2)392 1668 y Fk(Z)450 1736 y Fm(')482 1716 y Fh(0)482 1748 y Fl(T)510 1736 y Fm(uu)566 1743 y Fl(x)598 1736 y Fo(+)647 1668 y Fk(Z)705 1736 y Fm(x')765 1716 y Fh(00)765 1748 y Fl(T)793 1736 y Fm(uu)849 1743 y Fl(x)881 1736 y Fo(+)935 1702 y(3)p 935 1725 V 935 1770 a(2)973 1668 y Fk(Z)1031 1736 y Fm(x')1091 1716 y Fh(0)1091 1748 y Fl(T)1118 1736 y Fm(u)1146 1716 y Fi(2)1146 1748 y Fl(x)1179 1736 y Fo(+)k Fm(")1256 1702 y Fo(1)p 1256 1725 V 1256 1770 a(2)1293 1668 y Fk(Z)1352 1736 y Fm(x')1412 1716 y Fh(0)1412 1748 y Fl(T)1439 1736 y Fm(v)1465 1716 y Fi(2)1498 1736 y Fo(=)i(0)p Fm(:)164 1845 y Fo(Dividing)h(the)i(last)f (iden)o(tit)o(y)e(b)o(y)i Fm(T)7 b Fo(,)14 b(w)o(e)h(are)h(going)g(to)f (pass)i(to)e(the)g(limit)e(as)j Fm(T)k Fj(!)14 b(1)p Fo(.)164 1905 y(First,)h(w)o(e)h(p)q(oin)o(t)g(out)h(that)g(here)f(the) g(in)o(tegrals)f(con)o(taining)h Fm(')1343 1912 y Fl(T)1387 1905 y Fo(are)g(tak)o(en)g(o)o(v)o(er)164 2015 y Fm(Q)203 2022 y Fl(T)236 2015 y Fj([)6 b Fm(Q)314 1994 y Fh(0)314 2027 y Fl(T)347 2015 y Fj([)g Fm(Q)425 1994 y Fh(00)425 2027 y Fl(T)466 2015 y Fo(=)14 b Fm(Q)557 2022 y Fl(T)590 2015 y Fj([)6 b(f)p Fo(\()p Fm(T)f(=)p Fo(2)p Fm(;)j Fo(\()p Fm(T)13 b Fo(+)6 b Fm(T)918 1994 y Fl(\024)940 2015 y Fo(\))p Fm(=)p Fo(2\))g Fj(\002)g Ff(R)p Fj(g)g([)g(f)p Fo(\()p Fj(\000)p Fo(\()p Fm(T)j Fo(+)d Fm(T)1406 1994 y Fl(\024)1428 2015 y Fo(\))p Fm(=)p Fo(2)p Fm(;)i Fj(\000)p Fm(T)d(=)p Fo(2\))h Fj(\002)g Ff(R)p Fj(g)p Fm(;)164 2125 y Fo(while)16 b(that)h(ones)g(con)o(taining)g Fm(')777 2107 y Fh(0)777 2137 y Fl(T)821 2125 y Fo(and)g Fm(')948 2107 y Fh(00)948 2137 y Fl(T)992 2125 y Fo(are)g(o)o(v)o(er)f Fm(Q)1218 2107 y Fh(0)1218 2137 y Fl(T)1256 2125 y Fj([)c Fm(Q)1340 2107 y Fh(00)1340 2137 y Fl(T)1367 2125 y Fo(.)23 b(Moreo)o(v)o(er,)15 b Fm(')1664 2132 y Fl(T)1706 2125 y Fo(=)g(1)164 2185 y(on)i Fm(Q)271 2192 y Fl(T)298 2185 y Fo(.)237 2245 y(No)o(w)h(let)f Fm(g)i Fj(2)e Fm(L)547 2227 y Fi(1)547 2258 y Fl(loc)592 2245 y Fo(\()p Ff(R)650 2227 y Fi(2)667 2245 y Fo(\))h(b)q(e)g(a)g(function)g(whic)o(h)f(is)g Fm(k)r Fo(-p)q(erio)q(dic)h(in)g Fm(x)p Fo(.)26 b(Then,)18 b(it)f(is)164 2305 y(easy)f(to)h(v)o(erify)d(that)756 2387 y(lim)742 2418 y Fl(T)5 b Fh(!1)857 2353 y Fo(1)p 852 2375 36 2 v 852 2421 a Fm(T)900 2319 y Fk(Z)928 2432 y Fl(Q)956 2438 y Fg(T)990 2387 y Fm(g)16 b Fo(=)1081 2319 y Fk(Z)1109 2432 y Fl(Q)1137 2438 y Fg(k)1166 2387 y Fm(g)r(:)949 2607 y Fo(14)p eop %%Page: 15 16 15 15 bop 164 315 a Fo(Next,)23 b(due)g(to)g(the)g(prop)q(erties)f(of)h Fm(')890 322 y Fl(T)918 315 y Fo(,)h(all)e(the)h(in)o(tegrals)f(o)o(v)o (er)g Fm(Q)1474 297 y Fh(0)1474 328 y Fl(T)1524 315 y Fo(can)h(b)q(e)g(esti-)164 376 y(mated)f(from)g(ab)q(o)o(v)o(e)h(b)o(y) 662 336 y Fk(R)686 393 y Fl(Q)714 382 y Fa(0)714 405 y Fg(T)748 376 y Fm(g)r Fo(,)i(with)e(a)h(nonnegativ)o(e)f Fm(k)r Fo(-p)q(erio)q(dic)g(in)g Fm(x)g Fo(function)164 448 y Fm(g)16 b Fj(2)e Fm(L)283 430 y Fi(1)283 461 y Fl(loc)329 448 y Fo(\()p Ff(R)386 430 y Fi(2)403 448 y Fo(\).)21 b(No)o(w)724 499 y(1)p 719 522 36 2 v 719 567 a Fm(T)767 465 y Fk(Z)795 578 y Fl(Q)823 567 y Fa(0)823 590 y Fg(T)858 533 y Fm(g)15 b Fj(\024)954 499 y Fm(T)990 481 y Fl(\024)1023 499 y Fo(+)c(1)p 954 522 143 2 v 1007 567 a Fm(T)1110 465 y Fk(Z)1137 578 y Fl(Q)1165 584 y Fg(k)1194 533 y Fm(g)r(:)164 655 y Fo(This)16 b(justi\014es)g(the)g (passage)i(to)f(the)f(limit)d(and)k(giv)o(es)f(rise)f(to)i(\(15\).)237 715 y(Iden)o(tit)o(y)h(\(16\))i(can)g(b)q(e)g(pro)o(v)o(ed)f(exactly)g (as)h(\(2.8\),)h([5],)f(with)f(the)h(only)f(c)o(hange:)164 775 y(tak)o(e)d(there)f(cut-o\013)i(functions)g Fm(\037)789 782 y Fl(j)823 775 y Fo(dep)q(ending)f(on)h Fm(y)h Fo(only)l(.)237 835 y(No)o(w)e(w)o(e)g(construct)g(the)g(function)g Fm(')941 842 y Fl(T)969 835 y Fo(.)21 b(Fix)15 b Fm(")f(>)g Fo(0)j(and)f(let)353 993 y Fm(g)r Fo(\()p Fm(x)p Fo(\))e(=)510 891 y Fk(8)510 936 y(<)510 1026 y(:)575 933 y Fo(1)305 b(if)48 b Fm(x)13 b Fj(\024)h Fm(T)5 b(=)p Fo(2)575 993 y(1)11 b Fj(\000)g Fo(log)q(\()p Fm(x=T)c Fo(\))55 b(if)48 b Fm(T)5 b(=)p Fo(2)14 b Fm(<)g(x)g Fj(\024)f Fo(\()p Fm(T)18 b Fo(+)11 b Fm(T)1374 975 y Fl(\024)1396 993 y Fo(\))p Fm(=)p Fo(2)g(+)g Fm(")575 1053 y Fo(0)305 b(if)48 b Fm(x)13 b(>)h Fo(\()p Fm(T)k Fo(+)11 b Fm(T)1225 1035 y Fl(\024)1246 1053 y Fo(\))p Fm(=)p Fo(2)h(+)f Fm(")1580 993 y(:)164 1155 y Fo(W)l(e)16 b(c)o(ho)q(ose)g(a)g(nonnegativ)o(e)g(function)f Fm(h)f Fj(2)g Fm(C)1031 1136 y Fh(1)1027 1167 y Fi(0)1068 1155 y Fo(\()p Ff(R)p Fo(\))f(suc)o(h)i(that)i(supp)d Fm(h)g Fj(\032)f Fo(\(0)p Fm(;)8 b(")p Fo(\))16 b(and)164 1175 y Fk(R)205 1215 y Fm(h)e Fo(=)g(1,)i(and)h(set)471 1344 y(~)-32 b Fm(')495 1351 y Fl(T)522 1344 y Fo(\()p Fm(x)p Fo(\))14 b(=)654 1276 y Fk(Z)712 1344 y Fm(h)p Fo(\()p Fm(x)d Fj(\000)f Fm(t)p Fo(\))p Fm(g)r Fo(\()p Fm(t)p Fo(\))p Fm(dt;)105 b(')1159 1351 y Fl(T)1186 1344 y Fo(\()p Fm(x)p Fo(\))14 b(=)22 b(~)-32 b Fm(')1350 1351 y Fl(T)1377 1344 y Fo(\()p Fj(j)p Fm(x)p Fj(j)p Fo(\))p Fm(:)164 1467 y Fo(F)l(or)16 b(this)g(function)g(it)g(is)g (easy)h(to)f(v)o(erify)f(all)g(the)h(prop)q(erties)h(w)o(e)f(need.)k Fb(2)164 1572 y Fp(Theorem)d(5)24 b Fe(Supp)n(ose)18 b(that)g Fm(f)j Fj(2)15 b Fm(C)t Fo(\()p Ff(R)o Fo(\))g Fe(satis\014es)k(Assumption)f Fo(4)1439 1554 y Fi(0)1459 1572 y Fe(.)24 b(Then)19 b(ther)n(e)f(is)164 1632 y(no)f(nontrivial)i (tr)n(aveling)f(wave)g Fm(u)c Fj(2)g Fm(Y)891 1639 y Fl(k)913 1632 y Fe(,)j Fm(k)f Fj(2)e Fo(\(0)p Fm(;)8 b Fj(1)p Fo(])p Fe(,)17 b(pr)n(ovide)n(d)f Fm(")d Fo(=)h(+1)p Fe(,)k(or)e Fm(")e Fo(=)g Fj(\000)p Fo(1)164 1692 y Fe(and)k Fm(\026)c Fj(\025)f Fo(6)p Fe(.)237 1797 y(Pr)n(o)n(of)c Fo(.)21 b(Adding)16 b(\(15\),)h(\(16\))g(and)g(subtracting)f(\(17\),)h (w)o(e)f(get)754 1855 y Fk(Z)782 1968 y Fl(Q)810 1974 y Fg(k)839 1923 y Fm(u)867 1902 y Fi(2)867 1935 y Fl(x)903 1923 y Fo(=)e Fj(\000)p Fo(2)p Fm(")1049 1855 y Fk(Z)1077 1968 y Fl(Q)1105 1974 y Fg(k)1134 1923 y Fm(v)1160 1902 y Fi(2)1179 1923 y Fm(:)164 2058 y Fo(This)g(rules)g(out)h(the)f(case)h Fm(")e Fo(=)h(+1.)21 b(In)14 b(the)g(case)h Fm(")e Fo(=)h Fj(\000)p Fo(1)h(\(KP-I)f(equations\))h(the)f(last)164 2118 y(iden)o(tit)o(y)g(together)j(with)f(\(15\))h(and)f(\(17\),)h (resp)q(ectiv)o(ely)l(,)c(implies)592 2176 y Fk(Z)620 2289 y Fl(Q)648 2295 y Fg(k)669 2244 y Fo([)690 2210 y Fm(c)p 688 2233 25 2 v 688 2278 a Fo(2)716 2244 y Fm(u)744 2223 y Fi(2)775 2244 y Fo(+)829 2210 y(5)p 829 2233 V 829 2278 a(2)858 2244 y Fm(v)884 2223 y Fi(2)915 2244 y Fj(\000)d Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)11 b Fo(+)g Fm(F)c Fo(\()p Fm(u)p Fo(\)])13 b(=)g(0)164 2376 y(and)679 2380 y Fk(Z)707 2493 y Fl(Q)735 2499 y Fg(k)756 2448 y Fo([)p Fm(cu)819 2427 y Fi(2)849 2448 y Fo(+)e(3)p Fm(v)948 2427 y Fi(2)978 2448 y Fj(\000)g Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)p Fo(])13 b(=)h(0)p Fm(:)949 2607 y Fo(15)p eop %%Page: 16 17 16 16 bop 164 315 a Fo(Eliminating)14 b Fm(v)r Fo(,)h(w)o(e)h(get)633 447 y(2)p Fm(c)686 379 y Fk(Z)714 492 y Fl(Q)742 498 y Fg(k)771 447 y Fm(u)799 426 y Fi(2)833 447 y Fo(=)885 379 y Fk(Z)912 492 y Fl(Q)940 498 y Fg(k)961 447 y Fo([6)p Fm(F)7 b Fo(\()p Fm(u)p Fo(\))j Fj(\000)h Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)p Fo(])p Fm(:)164 587 y Fo(If)16 b Fm(\026)e Fj(\025)f Fo(6,)k(w)o(e)e(ha)o(v)o(e)585 718 y(2)p Fm(c)638 650 y Fk(Z)666 763 y Fl(Q)694 769 y Fg(k)723 718 y Fm(u)751 698 y Fi(2)785 718 y Fj(\024)837 650 y Fk(Z)865 763 y Fl(Q)893 769 y Fg(k)914 718 y Fo([)p Fm(\026F)7 b Fo(\()p Fm(u)p Fo(\))j Fj(\000)g Fm(f)5 b Fo(\()p Fm(u)p Fo(\))p Fm(u)p Fo(])14 b Fj(\024)f Fo(0)p Fm(:)164 858 y Fo(Hence,)h Fm(u)g Fo(=)g(0)i(and)h(w)o(e)f(conclude.)21 b Fb(2)237 1004 y Fp(Ac)n(kno)n(wledgemen)n(ts)p Fo(.)44 b(This)24 b(w)o(ork)g(w)o(as)h(carried)f(out)g(during)g(the)g(visit)g (of)164 1064 y(the)d(\014rst)h(author)g(\(A.P)l(.\))e(to)h(the)g (Institut)g(f)q(\177)-26 b(ur)22 b(Mathematik,)f(Hum)o(b)q(oldt)e(Univ) o(er-)164 1124 y(sit\177)-24 b(at)20 b(Berlin)f(\(Septem)o(b)q(er)f(|)i (Decem)o(b)q(er,)e(1998\))k(under)e(supp)q(ort)h(from)f(Deutsc)o(he)164 1184 y(F)l(orsc)o(h)o(ungsgemeinsc)o(haft.)e(A.P)l(.)10 b(is)i(v)o(ery)e(grateful)i(to)f(K.)h(Gr\177)-24 b(oger)12 b(for)g(his)f(kind)g(hospi-)164 1245 y(talit)o(y)h(and)j(a)f(lot)f(of)h (stim)o(ulating)e(discussions.)21 b(The)14 b(authors)h(thank)f(the)f (anonimous)164 1305 y(referee)i(for)h(information)f(on)i(the)f(pap)q (er)h([24].)164 1471 y Fn(References)188 1581 y Fo([1])24 b(Ablo)o(witz,)13 b(M.)g(J.,)g(Segur,)i(H.)d(and)j(W)l(ang,)g(X.)e(P)l (.,)g(W)l(a)o(v)o(e)g(collapse)h(and)g(instabil-)264 1641 y(it)o(y)j(of)h(solitary)g(w)o(a)o(v)o(es)f(of)h(a)g(generalized)f (Kadom)o(tsev-P)o(etviash)o(vili)d(equation,)264 1701 y Fe(Physic)n(a)j(D)5 b Fo(,)15 b Fp(78)p Fo(,)h(No.)g(3{4)h(\(1994\),) h(241{265.)188 1803 y([2])24 b(Bartsc)o(h,)14 b(Th.)g(and)g(Willem,)d (M.,)j(In\014nitely)e(man)o(y)g(radial)i(solutions)h(of)f(a)h(semi-)264 1863 y(linear)e(elliptic)f(problem)g(on)j Ff(R)847 1845 y Fl(N)877 1863 y Fo(,)f Fe(A)o(r)n(ch.)h(R)n(at.)f(Me)n(ch.)h(A)o (nal.)p Fo(,)g Fp(124)f Fo(\(1993\),)h(261{)264 1923 y(276.)188 2025 y([3])24 b(Bourgin,)h(J.,)f(On)f(the)g(Cauc)o(h)o(y)g (problem)f(for)h(the)g(Kadom)o(tsev-P)o(etviash)o(vili)264 2085 y(equation,)16 b Fe(Ge)n(om.)h(F)l(unct.)h(A)o(nal.)p Fo(,)f Fp(3)p Fo(,)f(No.)f(4)i(\(1993\),)g(315{341.)188 2187 y([4])24 b(De)16 b(Bouard,)g(A.,)e(Saut,)i(J.-C.,)f(Remarks)f(on)i (the)f(stabilit)o(y)g(of)g(generalized)g(KP)264 2247 y(solitary)h(w)o(a)o(v)o(es,)f Fe(Contemp.)j(Math.)p Fo(,)e Fp(200)g Fo(\(1996\),)h(75{84.)188 2349 y([5])24 b(De)c(Bouard,)g(A.,)f(Saut,)h(J.-C.,)g(Solitary)f(w)o(a)o(v)o(es)g(of) h(generalized)e(Kadom)o(tsev-)264 2409 y(P)o(etviash)o(vili)23 b(equations,)28 b Fe(A)o(nn.)e(Inst.)h(H.)f(Poinc)n(ar)o(\023)-24 b(e,)28 b(A)o(nal.)e(Non)h(Lin.)p Fo(,)g Fp(14)264 2469 y Fo(\(1997\),)18 b(211{236.)949 2607 y(16)p eop %%Page: 17 18 17 17 bop 188 315 a Fo([6])24 b(De)29 b(Bouard,)k(A.,)e(Saut,)i(J.-C.,) e(Symmetry)26 b(and)k(deca)o(y)e(of)i(the)f(general-)264 376 y(ized)17 b(Kadom)o(tsev-P)o(etviash)o(vili)e(solitary)j(w)o(a)o(v) o(es,)f Fe(SIAM)i(J.)g(Math.)g(A)o(nal.)p Fo(,)f Fp(28)p Fo(,)264 436 y(\(1997\),)g(1064{1085.)188 538 y([7])24 b(Kadom)o(tsev,)12 b(B.)f(B.)g(and)i(P)o(etviash)o(vili,)d(V.)i(I.,)g (On)g(stabilit)o(y)f(of)h(w)o(a)o(v)o(es)g(in)g(w)o(eakly)264 598 y(disp)q(ersiv)o(e)21 b(media,)f Fe(Sov.)j(Phys.)f(Dokl.)p Fo(,)h Fp(15)p Fo(,)f(\(1970\),)i(539{541,)h(transl.)c(from)264 658 y Fe(Dokl.)d(AN)g(SSSR)r Fo(,)f Fp(192)p Fo(,)e(\(1970\),)j (753{756.)188 760 y([8])24 b(Kric)o(hev)o(er,)13 b(I.)h(M.)g(and)i(No)o (vik)o(o)o(v)d(S.)h(P)l(.,)h(Holomorphic)e(bundles)i(o)o(v)o(er)f (algebraic)264 820 y(curv)o(es)20 b(and)h(nonlinear)g(equations,)g Fe(R)o(uss.)g(Math.)g(Surv.)p Fo(,)g Fp(35)p Fo(,)h(No.)e(6)h (\(1980\),)264 880 y(53{79,)d(transl.)e(from)f Fe(Usp.)j(Mat.)f(Nauk.)p Fo(,)f Fp(35)p Fo(,)g(No)g(6)h(\(1980\),)g(47{68.)188 982 y([9])24 b(Lions,)19 b(P)l(.-L.,)e(The)h(concen)o (tration-compactness)e(metho)q(d)h(in)g(the)h(calculus)f(of)264 1042 y(v)m(ariations.)c(The)f(lo)q(cally)g(compact)f(case.)h(I,)g(I)q (I,)g Fe(A)o(nn.)i(Inst.)g(H.)h(Poinc)n(ar)o(\023)-24 b(e,)14 b(A)o(nal.)264 1102 y(Non)k(Lin.)p Fo(,)e Fp(1)g Fo(\(1984\),)h(109{145,)i(223{283.)164 1204 y([10])24 b(Liu)f(Y)l(ue)g(and)h(W)l(ang,)h(X.)d(P)l(.,)i(Nonlinear)f(stabilit)o (y)e(of)j(solitary)f(w)o(a)o(v)o(es)f(of)i(a)264 1264 y(generalized)f(Kadom)o(tsev-P)o(etviash)o(vili)d(equation,)25 b Fe(Commun.)f(Math.)g(Phys.)p Fo(,)264 1324 y Fp(183)17 b Fo(\(1997\),)g(253{266.)164 1426 y([11])24 b(Lizorkin,)12 b(P)l(.)g(I.,)g(Multipliers)e(of)j(F)l(ourier)f(in)o(tegrals,)g Fe(Pr)n(o)n(c.)h(Steklov)j(Inst.)e(Math.)p Fo(,)264 1486 y Fp(89)j Fo(\(1967\),)g(269{290.)164 1588 y([12])24 b(Lop)q(es,)f(O.,)e(A)f(constrained)h(minim)o(iz)o(ation)d(problem)h (with)i(in)o(tegrals)f(on)h(the)264 1648 y(en)o(tier)15 b(space,)h Fe(Bol.)i(Sc.)g(Br)n(as.)f(Mat.)p Fo(,)e Fp(25)i Fo(\(1994\),)g(77{92.)164 1750 y([13])24 b(Nehari,)18 b(Z.,)h(On)f(a)h(class)g(of)g(nonlinear)g(second-order)g(di\013eren)o (tial)e(equations,)264 1810 y Fe(T)l(r)n(ans.)g(A)o(mer.)g(Math.)g(So)n (c.)p Fo(,)f Fp(95)h Fo(\(1960\),)g(101{123.)164 1912 y([14])24 b(P)o(ank)o(o)o(v,)19 b(A.)g(A.,)g(Semilinear)d(elliptic)i (equations)h(in)g Ff(R)1333 1894 y Fl(n)1373 1912 y Fo(with)g (nonstabilizing)264 1972 y(co)q(e\016cien)o(ts,)j Fe(Ukr.)h(Math.)f(J.) p Fo(,)h Fp(41)p Fo(,)g(No)f(9)h(\(1989\),)h(1075{107)q(8,)i(transl.)c (from)264 2032 y Fe(Ukr.)c(Mat.)f(Zh.)p Fo(,)e Fp(41)p Fo(,)h(No)h(9)f(\(1989\),)h(1247{125)q(1.)164 2134 y([15])24 b(P)o(ank)o(o)o(v,)18 b(A.)f(A.,)h(On)g(p)q(ositiv)o(e)g(solutions)g (of)h(nonlinear)f(elliptic)e(equations)i(on)264 2194 y(whole)g(space,)f Fe(Sov.)j(Math.,)e(Dokl.)p Fo(,)g Fp(44)g Fo(\(1991\),)g(337{341,)j(transl.)c(from)f Fe(Dokl.)264 2254 y(AN)i(SSSR)r Fo(,)f Fp(319)p Fo(,)f(No.)f(6)i(\(1991\),)g (1318{1321)q(.)164 2356 y([16])24 b(P)o(ank)o(o)o(v,)c(A.)f(A.)g(and)h (P\015)q(\177)-26 b(uger,)22 b(K.,)e(On)g(a)g(semilinear)d(Sc)o(hr\177) -24 b(odinger)20 b(equation)264 2416 y(with)13 b(p)q(erio)q(dic)h(p)q (oten)o(tial,)f Fe(Nonlin.)j(A)o(nal.,)g(The)n(ory,)e(Meth.,)h(Appl.)p Fo(,)f Fp(33)g Fo(\(1998\),)264 2476 y(593{609.)949 2607 y(17)p eop %%Page: 18 19 18 18 bop 164 315 a Fo([17])24 b(P)o(ank)o(o)o(v,)e(A.)f(A.)f(and)j (P\015)q(\177)-26 b(uger,)23 b(K.,)f(P)o(erio)q(dic)f(and)h(solitary)g (tra)o(v)o(eling)e(w)o(a)o(v)o(es)264 376 y(for)14 b(the)f(generalized) f(Kadom)o(tsev-P)o(etviash)o(vili)e(equation,)k Fe(Math.)g(Meth.)h (Appl.)264 436 y(Sci.)p Fo(,)i Fp(22)f Fo(\(1999\),)h(733{752.)164 538 y([18])24 b(P)o(ouget,)16 b(J.,)e(Stabilit)o(y)g(of)i(nonlinear)f (structures)h(in)f(a)g(lattice)g(mo)q(del)f(for)h(phase)264 598 y(transformations)i(in)f(allo)o(ys,)f Fe(Phys.)i(R)n(ev.)g(B.)p Fo(,)f Fp(46)g Fo(\(1992\),)i(10554{1056)q(2.)164 699 y([19])24 b(Rabino)o(witz,)15 b(P)l(.)h(H.,)e Fe(Minimax)j(metho)n(ds)g (in)g(critic)n(al)h(p)n(oint)f(the)n(ory)f(with)h(appli-)264 760 y(c)n(ations)h(to)f(di\013er)n(ential)h(e)n(quations)t Fo(,)f(AMS)f(Reg.)f(Conf.)i(Ser.)e(Math.)h Fp(65)p Fo(,)g(1986.)164 861 y([20])24 b(Saut,)19 b(J.-C.,)g(Remarks)e(on)i(the)f(generalized)g (Kadom)o(tsev-P)o(etviash)o(vili)d(equa-)264 922 y(tion,)h Fe(Indiana)i(Univ.)g(Math.)f(J.)p Fo(,)e Fp(42)i Fo(\(1993\),)g (1011{1026)q(.)164 1023 y([21])24 b(Stein,)15 b(E.)g(M.,)f(W)l(eiss,)h (G.,)f Fe(Intr)n(o)n(duction)j(to)f(F)l(ourier)g(A)o(nalysis)h(on)g (Eucle)n(adian)264 1083 y(Sp)n(ac)n(es)t Fo(,)f(Princeton)g(Univ.)f (Press,)h(Princeton,)f(1971.)164 1185 y([22])24 b(Willem,)13 b(M.,)i Fe(Minimax)j(Metho)n(ds)t Fo(,)d(Birkh\177)-24 b(auser,)15 b(Boston)i(etc.,)e(1996.)164 1287 y([23])24 b(Willem,)10 b(M.,)i(On)h(the)f(generalized)f(Kadom)o(tsev-P)o(etviash) o(vili)e(equation,)k Fe(R)n(app.)264 1347 y(Semin.)19 b(Math.)e(L)n(ouvain,)g(Nov.)h(Ser.)p Fo(,)e Fp(245{260)g Fo(\(1996\),)i(213{222.)164 1449 y([24])24 b(Zaitsev,)14 b(A.)h(A.,)f(On)h(formation)f(of)i(nonlinear)f(stationary)g(w)o(a)o(v)o (es)g(b)o(y)f(means)h(of)264 1509 y(sup)q(erp)q(osition)h(of)g (solitons,)f Fe(Dokl.)i(A)o(kad.)f(Nauk)h(SSSR)r Fo(,)f Fp(272)f Fo(\(1983\),)h(583{587)264 1569 y(\(in)g(Russian\).)949 2607 y(18)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0004200312884--