Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions

Using he ini e domain emnan o he con inuous spec um o examine in eg abili y: E ec o bounda y condi ions Panayo is G. Ke ekidis1and Niu ka R. Quin e o2 1Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , Massachuse s 01003-4515, USA 2Depa amen o de Fı ´sica Aplicada I, Escuela Uni e si a ia Poli e ´cnica, Uni e sidad de Se illa, Vi gen de A ´ ica 7, 41011 Se illa, Spain and Ins i u o Ca los I de Fı ´sica Teo ´ ica y Compu acional, Uni e sidad de G anada, E-18071 G anada, Spain 共Recei ed 11 Decembe 2002; e ised manusc ip ecei ed 30 June 2003; published 23 Sep embe 2003兲 The aim o his wo k is o p opose a me hod o es ing he in eg abili y o a model pa ial di e en ial 共PDE兲 and/o di e en ial di e ence equa ion 共DDE兲, by examining i in a ini e bu la ge domain. Fo monopa ame - ic amilies o PDE/DDE’s, ha a e known o possess isola ed in eg able poin s, we ind ha e y special ea u es occu in he ini e domain emnan o he con inuous 共‘‘phonon’’兲spec um a hese ‘‘singula ’’ poin s. We iden i y hese ea u es in he case example o a PDE and a DDE 共 ha sus ain on and pulselike solu ions, espec i ely兲 o di e en ypes o bounda y condi ions. The key inding o he wo k is ha such spec al ea u es a e gene ic nea he singula , in eg able poin s and hence we p opose o explo e a gi en PDE/DDE in a ini e bu la ge domain o such ai s, as a means o assessing i s po en ial in eg abili y. DOI: 10.1103/PhysRe E.68.036612 PACS numbe 共s兲: 05.45.⫺a, 02.30.Ik I. INTRODUCTION In eg able models o pa ial di e en ial 共PDE兲and di e - en ial di e ence 共DDE兲equa ions ha e been a opic o in- ense in es iga ion o e he pas ew decades 关1–3兴. The main eason o his, excep o he wide a ie y o physical applica ions ha can be desc ibed by in eg able o nea - in eg able sys ems, is ha he special case o in eg able mod- els can be analyzed comple ely by means o he in e se sca - e ing ans o m 关1,4兴. This can hen se e as a s a ing poin o pe u ba i e ea men o nea -in eg able sys ems. In he p ocess o hese de elopmen s, a numbe o ech- niques ha e been de eloped o assessing in eg abili y in con inuous 关5兴o disc e e 关6兴se ings 共o applicable o bo h 关7兴兲. An in e es ing ea u e o hese ‘‘ es s’’ is ha hey a e necessa y 共bu no su icien 兲condi ions o in eg abili y. Hence, i a model equa ion ails such a c i e ion, i is nonin- eg able, bu i i passes, i may o may no be in eg able. In a sense, his sugges s ha we s ill do no unde s and he essen ial ing edien s ha ende a sys em comple ely in e- g able. O cou se, should a Lax pai be iden i ied and he in e se sca e ing mechanism be applied, we know ha he sys em is in eg able, bu i would ce ainly be desi able 共as is clea om all he abo e e o o c ea e ‘‘in eg abili y es s’’兲 o ha e a mechanis ic 共‘‘black box’’兲 ype o c i e ion o assess ha . We, o cou se, do no claim o be p o iding a ull answe o his ques ion in he p esen wo k. Howe e , we will y o gi e a numbe o use ul hin s ha may lead o pa ial an- swe s o he abo e ques ions and may p o ide some in ui ion in he e o o cons uc such mechanis ic c i e ia. Ou ool o choice will be he use o di e en se s o bounda y condi ions 共BC兲 o examine he spec um o he linea iza ion a ound he nonlinea cohe en s uc u e ha he PDE/DDE o in e es suppo s. No ice ha he e ec o bounda y condi ions in ela ed con ex s has been s udied in a numbe o e e ences; see, e.g., Re . 关8兴, and e e ences he ein. Howe e , in all o hese wo ks he e ec s o he BC o he poin spec um we e assessed and mo eo e , his was no done in di ec connec ion wi h issues o in eg abili y. He e we will, ins ead, ocus on he con inuous spec um; in ac , since we will be dealing wi h ini e bu la ge domains, we will cen e ou a en ion a ound he disc e e spec um emnan ha ‘‘becomes’’ he con inuous spec um in he in- ini e domain limi . In he ini e domain case, he 共 o me ly con inuous兲spec um becomes disc e e due o he quan iza- ion o he wa e numbe s, imposed by he bounda y condi- ions 共see, e.g., Sec. II below兲. I is exac ly his disc e e em- nan o he con inuous spec um, ha we aim a examining he e, o elucida e i s in e es ing p ope ies in in eg able e - sus nonin eg able se ings. In he p esen wo k, we ocus on wo model p oblems, o es ablish ou indings and demons a e hei gene ali y. The models a e selec ed as one-pa ame e amilies o equa ions such ha one membe o he amily is an in eg able sys em. Mo eo e , in illus a ing he gene ali y o he conclusions, hey a e selec ed in a o m such ha one model co esponds o a PDE, while he o he o a DDE, so ha one is kink bea ing, while he o he is pulse bea ing. The models o in- e es will be he pa ame ically modi ied sine-Go don equa- ion 关o en also called he Pey a d-Remoissene 共PR兲model兴 关9,10兴and a modi ied e sion o he disc e e nonlinea Sch o ¨dinge 共DNLS兲model 共occasionally called he Sale no model兲关11兴. The o me PDE eads ␾ ⫺ ␾ xx⫽⫺ dU d ␾ ,U共 ␾ , 兲⫽共1⫺ 兲2关1⫺cos共 ␾ 兲兴 1⫹ 2⫹2 cos共 ␾ 兲 共1兲 in he in ini e domain 兩 x 兩 ⬍⬁and wi h 兩 兩 ⬍1; while he la e DDE is o he o m iu ˙n⫽⫺⌬2un⫺ 兩 un 兩 2关2 ⑀ un⫹共1⫺ ⑀ 兲共un⫹1⫹un⫺1兲兴.共2兲 The mos well known among hese monopa ame ic amilies o models a e he sine-Go don equa ion 关Eq. 共1兲, o ⫽0] which is ele an o supe conduc i i y and cha ge densi y PHYSICAL REVIEW E 68, 036612 共2003兲 1063-651X/2003/68共3兲/036612共7兲/$20.00 ©2003 The Ame ican Physical Socie y68 036612-1 wa es among o he applica ions 关2兴and he expe imen ally ealizable disc e e nonlinea Sch o ¨dinge equa ion 关12兴o ⑀ ⫽1, as well as i s in eg able, so-called Ablowi z-Ladik 关13兴 coun e pa o ⑀ ⫽0 in he case o Eq. 共2兲. No ice ha o he PDE, he subsc ip s deno e pa ial de- i a i es o he ield, while o he DDE, he o e do deno es empo al de i a i e, ⌬2un⬅C(un⫹1⫺2un⫹un⫺1), whe e C⫽1/(⌬x)2is a cons an de e mined by he la ice spacing ⌬x; he subsc ip ndeno es he la ice si e index. In he o me case, he e exis kinklike solu ions which ha e been de ailed in Re s. 关9,10兴, while in he la e , he ield is com- plex and he e exis pulselike solu ions o he o m un ⫽exp(i⌳ ) n, whe e ⌳is he equency o he solu ions and ni s 共 eal兲exponen ially localized spa ial p o ile 关11,12兴. In he PDE, linea iza ion a ound a s a e ␾ 0(x), using he ansa z ␾ ⫽ ␾ 0(x)⫹ ␦ exp(i ␻ ) (x) in o Eq. 共1兲, yields o O( ␦ ) he linea iza ion equa ion xx⫹关 ␻ 2⫺U⬙共 ␾ 0, 兲兴 ⫽0. 共3兲 No ice ha when ⫽0共in he in ini e domain limi 兲, ␾ 0(x) ⫽4 a c an关exp(x)兴is he s a ic kink solu ion o he sG equa- ion and o his unc ion, he S u m-Liou ille p oblem 共3兲 can be exac ly sol ed 关14兴yielding one disc e e mode 共Gold- s one mode兲a ␻ ⫽0 and he con inuous spec um ep e- sen ed by he phonons, ␻ k⫽ 冑 1⫹k2, k共x兲⫽exp共ikx兲 冑 2 ␲␻ k 关k⫹i anh共x兲兴,共4兲 o all alues o k. Fo ⫽0, nei he he s a ic solu ion no he linea iza ion spec um a e explici ly a ailable in he in i- ni e domain limi . Analogously o he PDE, o he linea s abili y analysis o DDE 共2兲we inse exp(i⌳ )关 n⫹ ␦ (Une⫺i ␻ ⫹Wnei ␻ 쐓 )兴in o Eq. 共2兲. We hus ob ain o O( ␦ ) he ollowing eigen alue p oblem o 兵 ␻ , 兵 Un,Wn 쐓 其 其: ␻ 冉 Un Wn 쐓 冊 ⫽L 冉 Un Wn 쐓 冊 ,L⫽ 冉 AB ⫺B⫺A 冊 , Amn⫽关⌳⫹2C⫺ 兵 4 ⑀ n 2⫹共1⫺ ⑀ 兲 n关 n⫹1⫹ n⫺1兴 其 兴 ␦ m,n ⫹关共1⫺ ⑀ 兲 n 2⫺C兴共 ␦ m,n⫹1⫹ ␦ m,n⫺1兲, Bmn⫽⫺ n关2 ⑀ n⫹共1⫺ ⑀ 兲共 n⫹1⫹ n⫺1兲兴 ␦ m,n,共5兲 whe e he s a s deno e complex conjuga ion. The pape is o ganized as ollow. In he ollowing sec ion we ob ain an app oxima e solu ion o he S u m-Liou ille p oblem 共3兲by imposing di e en ypes o bounda y condi- ions in he ini e domain o leng h L. The ob ained esul s a e compa ed wi h he nume ical compu a ions in Sec. III, whe e we also compu ed he solu ion o Eq. 共5兲. Finally, we summa ize ou indings and p esen ou conclusions in Sec. IV. II. ANALYTICAL APPROXIMATION In his sec ion we sol e app oxima ely Eq. 共3兲when 兩 x 兩 ⬍L/2, whe e Lis he ini e 共bu la ge enough兲leng h o he sys em. No ice ha ou esul s will be gene ically ue, i Lis chosen la ge enough. By la ge enough he e, we mean a do- main size which is many imes 共a leas 10兲la ge han he cha ac e is ic leng h o he soli a y wa e 共kink o pulse兲 ha we will examine inside his domain. We will ake in o ac- coun di e en kinds o BC, in pa icula , ee x共⫺L/2兲⫽0, x共L/2兲⫽0, 共6兲 ixed 共⫺L/2兲⫽0, 共L/2兲⫽0, 共7兲 and an ipe iodic bounda y condi ion 共aPBC兲 x共⫺L/2兲⫽⫺ x共L/2兲, 共⫺L/2兲⫽⫺ 共L/2兲.共8兲 Fi s we conside he in eg able case, ⫽0, and we show ha o he i s phonon modes, he eigen equencies ␻ ˜ n ee ⫽ ␻ ˜ n⫺1 ixed and ␻ ˜ n ap ha e a double mul iplici y 共we will deno e wi h ilde he analy ical, app oxima ed eigen equencies兲.To p oceed, we use he exac solu ion o p oblem 共3兲 o ⫽0in he in ini e domain. We would like o s ess ha i we change he in ini e domain by a ini e one, wi h a gi en BC, we will s ill ha e an in ini e numbe o eigen equencies,1bu o he allowed wa e numbe s k关15兴. In o de o calcula e app oxi- ma ely hese allowed wa e numbe s, we p oceed as in Re . 关16兴. No ice ha k(x)⫽Fk(x)⫹iGk(x), whe e Fk共x兲⫽kcos共kx兲⫺sin共kx兲 anh共x兲 冑 2 ␲␻ k ,共9兲 Gk共x兲⫽cos共kx兲 anh共x兲⫹ksin共kx兲 冑 2 ␲␻ k .共10兲 Then he solu ion o Eq. 共3兲, wi h ⫽0, ela ed o he pho- non con ibu ion is ep esen ed by he linea supe posi ion o all he odd 关Gk(x)兴and e en 关Fk(x)兴phonon modes ␺ 共x, 兲⫽兺 k关ak共 兲Fk共x兲⫹bk共 兲Gk共x兲兴.共11兲 Imposing ee BC o each phonon mode o Eq. 共11兲we ob ain ha he i s wa e numbe s sa is y ak共 兲关sin共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫹kcos共kL/2兲 anh共L/2兲兴 ⫽0, 共12兲 bk共 兲关cos共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫺ksin共kL/2兲 anh共L/2兲兴 ⫽0. 共13兲 1No e ha his is ue o he con inuum p oblem o Eq. 共3兲, bu would no longe be ue o he disc e e one o Eq. 共5兲. P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-2 The solu ions o hese anscenden al equa ions yield he al- lowed alues o k. We can sol e hese app oxima ely i we conside LⰇ1. Then, we ind ha kn,0 ee⫽n⫺1 L ␲ ,n⫽1,2,3,..., nⰆL,共14兲 whe e he ze o subsc ip deno es ha we a e dealing wi h he unpe u bed case ⫽0, and i s co esponding eigen unc ions a e ela ed wi h he odd unc ions Gn(x) o he odd numbe s nand wi h he e en unc ions Fn(x) o he e en numbe s n. Hence, he i s eigen equencies a e ep esen ed by ␻ ˜ n,0 ee⫽ 冑 1⫹ 冉 n⫺1 L ␲ 冊 2 ,n⫽1,2,..., nⰆL. 共15兲 Analogously, o ixed BC he ollowing ela ions hold; ak共 兲关kcos共kL/2兲⫺sin共kL/2兲 anh共L/2兲兴⫽0, 共16兲 bk共 兲关cos共kL/2兲 anh共L/2兲⫹ksin共kL/2兲兴⫽0. 共17兲 Then, o la ge enough L, we ind ha kn,0 ixed⫽n L ␲ ,n⫽1,2,3,..., nⰆL,共18兲 and so, ␻ ˜ n,0 ixed⫽ 冑 1⫹ 冉 n L ␲ 冊 2 ,n⫽1,2,..., nⰆL,共19兲 whe e he odd 共e en兲numbe s na e ela ed wi h he odd Gn(x)关e en Fn(x)] eigen unc ions. Rema k 1. By compa ing exp essions 共15兲and 共19兲we obse e ha in he in eg able case ( ⫽0) ␻ ˜ n ee⫽ ␻ ˜ n⫺1 ixed o he i s ew eigen equencies. Now by imposing aPBC in each phonon mode o Eq. 共11兲 and aking in o accoun he symme y p ope ies o Fk(x), Gk(x) and hei de i a i es, he equa ions ha he wa e numbe sa is ies can be educed o akFk共L/2兲⫽0, 共20兲 bk ⳵ Gk ⳵ x共L/2兲⫽0. 共21兲 No ice ha Eqs. 共20兲and 共21兲coincide wi h Eqs. 共16兲共i.e., he i s equa ion o ixed BC兲and 共13兲共i.e., he second equa ion o ee BC兲, espec i ely. The solu ions o Eqs. 共20兲and 共21兲a e gi en by kn,0 ap⫽2共n⫺1兲 ␲ L,n⫽2,3,..., nⰆL,共22兲 kn,0 ap⫽2共n⫺1兲 ␲ L,n⫽1,2,3,..., nⰆL,共23兲 espec i ely, and hei eigen unc ions co espond o he e en Fn(x) and odd Gn(x). Then, he i s eigen equencies a e ep esen ed by ␻ ˜ n,0 ap⫽ 冑 1⫹ 冉 2共n⫺1兲 ␲ L 冊 2 ,n⫽2,3..., nⰆL,共24兲 ␻ ˜ n,0 ap⫽ 冑 1⫹ 冉 2共n⫺1兲 ␲ L 冊 2 ,n⫽1,2,..., nⰆL. 共25兲 This means ha he e en 共odd兲modes o aPBC 兵 ␻ ˜ n ap ,an( )Fn(x) 其 关 兵 ␻ ˜ n ap ,bn( )Gn(x) 其 兴coincide wi h he e en modes o ixed BC 共odd modes o ee BC兲. Rema k 2. F om ela ions 共24兲and 共25兲we conclude ha o he in eg able case and aPBC he eigen equencies ha e mul iplici y 2. The analysis o he S u m-Liou ille p oblem 共3兲 o he nonin eg able case, ⫽0, becomes mo e complica ed since ␾ 0(x) is he exac kink solu ion o Eq. 共1兲and his unc ion is only known in he implici o m 关9兴共e en o he in ini e domain p oblem兲. So, ins ead o sol ing his equa ion we calcula e app oxima ely he solu ion o 冋 d2 dx2⫺V共x兲⫺ W共x兲⫹E 册 ⫽0, 共26兲 whe e V(x)⫽⫺2/cosh2(x), W(x)⫽8 anh(x)关x ⫺5 anh(x)兴/cosh2(x), and E⫽ ␻ 2⫺ ␻ ph 2wi h ␻ ph⫽(1 ⫺ )/(1⫹ )关10,17兴. This eigen alue p oblem is ob ained in wo s eps: i s we ind a solu ion o small o Eq. 共1兲, h ough he pe u ba i e expansion ␾ (x, )⫽ ␾ sG(x) ⫹ ␾ 1(x)⫹O( 2), whe e ␾ sG(x) is he s a ic sG kink and second we linea ize Eq. 共1兲a ound he ob ained solu ion up o o de o , so we inse ␾ (x, )⫽ ␾ sG(x)⫹ ␾ 1(x) ⫹ ␦ 关 (x)exp(i ␻ )⫹ 쐓(x)exp(⫺i ␻ )兴in o Eq. 共1兲and conside he equa ion ha a ises o O( ␦ ) and ob ain Eq. 共26兲. A gu- ably, his app oach ails o cap u e he co ec ions o he ail o he wa e due o domain ini eness. Howe e , as a gued in Re . 关8兴, he la e a e exponen ially small in he leng h o he domain. Hence, as will also be jus i ied a pos e io i, he e we cap u e he leading o de dependence in L, as well as he leading o de e ec o 关see, e.g., Eqs. 共31兲–共33兲below兴. Then, ollowing he p ocedu e o he pe u ba ion me hods o linea eigen alue p oblem sugges ed in Re . 关18兴,weas- sume he solu ion o Eq. 共26兲as En⫽En,0⫹ En,1⫹O共 2兲,共27兲 n共x兲⫽ n,0共x兲⫹ n,1共x兲⫹O共 2兲,共28兲 whe e he i s subsc ip in he unc ions, n, deno es he o de o he phonon modes 共 o ⬍0 his subsc ip can also deno e he in e nal mode兲, he second one co esponds o he o de o pe u ba ion. By inse ing hese expansions in Eq. 共26兲 and equa ing and collec ing he e ms o he same o de in , we ob ain o O( 0), USING THE FINITE DOMAIN REMNANT OF THE... PHYSICAL REVIEW E 68, 036612 共2003兲 036612-3 冋 d2 dx2⫺V共x兲⫹En,0 册 n,0⫽0, 共29兲 and o he nex o de co ec ion O( 1), 冋 d2 dx2⫺V共x兲⫹En,0 册 n,1⫽关W共x兲⫺En,1兴 n,0 .共30兲 No ice ha Eq. 共29兲co esponds o he in eg able case ⫽0 al eady sol ed o ee 关Eq. 共15兲兴, ixed 关Eq. 共19兲兴, and an ipe iodic BC 关see Eqs. 共24兲and 共25兲兴. No ice also ha En,0⫽ ␻ ˜ n,0 2⫺1⫽kn,0 2and ha i s co esponding eigen unc ion n,0(x) is ela ed ei he wi h he odd Gn(x) o e en Fn(x). Then, o di e en bounda y condi ions, he eigen equencies o Eq. 共26兲a e de e mined by ␻ ˜ n ee⫽ 冑 ␻ ph 2⫹共kn,0 ee兲2⫹ En,1 ee,共31兲 ␻ ˜ n ixed⫽ 冑 ␻ ph 2⫹共kn,0 ixed兲2⫹ En,1 ixed,共32兲 ␻ ˜ n ap⫽ 冑 ␻ ph 2⫹共kn,0 ap兲2⫹ En,1 ap,共33兲 whe e kn,0 ee ,kn,0 ixed , and kn,0 ap a e gi en by Eqs. 共14兲,共18兲, and 共22兲and 共23兲, espec i ely. The solu ion o he eigen alue En,1 o he i s -o de co - ec ion is gi en by En,1⫽ 冕 ⫺L/2 L/2 dx n,0共x兲W共x兲 n,0共x兲 冕 ⫺L/2 L/2 dx n,0 2共x兲 .共34兲 The in eg als in ol ed in Eq. 共34兲can be compu ed nume i- cally o di e en BC and di e en alues o ( 兩 兩 Ⰶ1), hen we can calcula e he app oxima ed eigen equencies in each case. We can now compa e hese esul s wi h he nume ical solu ions o Eq. 共3兲关 o de ails on he nume ical me hods/ esul s, we e e he eade o Sec. III兴. F om he da a o he Tables I and II we obse e an oscil- la o y beha io o ␻ ˜ n ee⫺ ␻ ˜ n⫺1 ixed o he i s phonon’s modes o ⫽0. We also no ice ha he eigen alues o aPBC lose hei double mul iplici y ha exis ed in he case o he in e- g able equa ion. I is also wo h no ing ha hese ea u es a e ypically obse able in he hi d decimal digi o he co esponding eigen equencies. On he o he hand, he di e ence 共well jus- i ied wi hin he app oxima ions men ioned abo e兲be ween he heo e ical and nume ical p edic ions o he indi idual eigen equencies is ypically in he ou h o i h decimal digi . Hence, he obse a ions o he p e ious pa ag aph a e sys ema ic and in ag eemen wi h he heo e ical p edic ions. III. NUMERICAL RESULTS AND DISCUSSIONS To ind he nume ical solu ion o Eqs. 共1兲and 共3兲,we disc e ize he equa ions in a nume ical mesh o a ini e do- main. The mesh consis s o he N⫹1 poin s xj⫽ 兵 ⫺L/2 ⫹j⌬x,j⫽0,1,2,...,N 其 de ined in he ini e leng h Lo he sys em (⌬x⫽L/N). No ice ha since, in his case, we wish o emula e he beha io o he PDE, ⌬xis e y ine 共 ypi- cally 0.05), and he obus ness o he indings upon a ia ion o he 共small兲⌬xhas been e i ied. When we compu e he solu ion ei he o he PDE o o he linea iza ion equa ion, we conside h ee di e en ypes o BC 共6兲–共8兲. We would like o ema k ha his kind o disc e iza ion o he S u m- Liou ille p oblem 共3兲only a ec s he las phonon modes, so we can compa e he beha io o he i s phonon modes ob- ained in he p eceding sec ion wi h he nume ical solu ion o Eq. 共3兲. In bo h cases, he dis ibu ions o hese eigen e- quencies a e de e mined by he pa ame e and by he di e en bounda y condi ions in he ini e domain. TABLE I. Fo posi i e and small alue o ⫽0.02, we compa e he i s eigen equencies, ob ained pe u ba i ely, ␻ ˜ n, wi h he ones compu ed by sol ing he o iginal Eqs. 共1兲and 共3兲, ␻ n. n ␻ n ee ␻ ˜ n ee ␻ n ixed ␻ ˜ n ixed ␻ n ap ␻ ˜ n ap 1 0.96117 0.96192 0.96234 0.96352 0.96117 0.96192 2 0.96132 0.96129 0.96293 0.96284 0.96293 0.96284 3 0.96436 0.96514 0.96729 0.96760 0.96436 0.96514 4 0.96560 0.96541 0.96933 0.96899 0.96933 0.96899 5 0.97118 0.97126 0.97604 0.97573 0.97118 0.97126 6 0.97411 0.97358 0.97989 0.97915 0.97991 0.97915 TABLE II. We p o ide he same compa ison as in he p e ious able o a nega i e alue o ⫽ ⫺0.02. He e, ⍀ ˜ iand ⍀i ep esen he in e nal mode calcula ed by he pe u ba ion me hod and compu ed by nume ical solu ion o Eq. 共3兲, espec i ely. n ␻ n ee ␻ ˜ n ee ␻ n ixed ␻ ˜ n ixed ␻ n ap ␻ ˜ n ap ⍀i⫽1.03560 ⍀ ˜ i⫽1.03977 ⍀i⫽1.03560 ⍀ ˜ i⫽1.03924 ⍀i⫽1.03560 ⍀ ˜ i⫽1.03977 1 1.04131 1.04129 1.04278 1.04271 1.04156 1.04058 2 1.04156 1.04058 1.04367 1.04303 1.04278 1.04271 3 1.04524 1.04506 1.04866 1.04835 1.04694 1.04625 4 1.04694 1.04625 1.05125 1.05057 1.04867 1.04835 5 1.05306 1.05257 1.05838 1.05770 1.05657 1.05564 P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-4 We also compu e he solu ions o DDE 共2兲and Eq. 共5兲 using 200 poin s and ⌬x⫽0.75. The BC a e de ined analo- gously h ough U0⫽U1,UN⫽UN⫺1,W0⫽W1, and WN ⫽WN⫺1 o ee BC. Fo ixed BC: U0⫽0, UN⫽0, W0 ⫽0, and WN⫽0, while o pe iodic BC: U0⫽UN⫺1,UN ⫽U1,W0⫽WN⫺1, and WN⫽W1. Ou esul s when he pa ame e o he PR po en ial o ⑀ in he DDE a e a ied can be summa ized in Figs. 1–6. F om he abo e esul s, he ollowing conclusions can be d awn. 共1兲Fo ixed BC, he band edge equency is p ohibi ed. Hence, we compa e ␻ n ee wi h ␻ n⫺1 ixed . We ind ha o small wa e numbe s, ixed and ee BC eigen equencies p ac i- cally coincide only in he in eg able case, whe eas o he nonin eg able case we obse e an oscilla o y beha io o his unc ion 关see Figs. 1 and 2兴. In Fig. 2 we also show he oscilla o y beha io o ␻ ˜ n ee⫺ ␻ ˜ n⫺1 ixed , ob ained om he pe - u ba ion heo y, o ⫽⫺0.02 共open iangles兲and ⫽0.02 共open squa es兲. 共2兲Fo an ipe iodic BC, he spec um comp ises o modes coming al e na ely om he ee and ixed BC. This seems na u al as he ee bounda y condi ions selec eigenmodes FIG. 1. Compa ison o he eigen equencies o he 共disc e e emnan o he兲con inuous spec um o ixed and ee BC: We ha e plo ed he di e ence be ween he eigen equencies compu ed om Eq. 共3兲and ␻ ph⫽(1⫺ )/(1⫹ ) s . The ci cles joined by solid line 共 ee BC兲 ep esen how a he equencies a e om he lowe phonon mode. The iangles joined by do ed lines co espond o ixed BC. FIG. 2. The di e ence be ween he i s equencies o ee and ixed BC, ␻ n ee⫺ ␻ n⫺1 ixed (2⭐n⭐20), is plo ed as a unc ion o he wa e numbe o ⫽0共ci cles joined by solid line兲, ⫽⫺0.02 共 i- angles joined by dashed line兲, and ⫽0.02 共squa es joined by do ed line兲. The open iangles ( ⫽⫺0.02) and squa es ( ⫽0.02) ep e- sen he di e ences be ween he equencies ob ained by he pe u - ba ion heo y, ␻ ˜ n ee⫺ ␻ ˜ n⫺1 ixed , in he p eceding sec ion. FIG. 3. An ipe iodic BC: he di e ence be ween he nume ical eigen equencies compu ed om Eq. 共3兲and he band edge o he 共 o me ly con inuous兲spec um ␻ ph⫽(1⫺ )/(1⫹ ) is shown. Ad- jacen eigenmodes a e gi en by ci cles joined by solid line and iangles joined by do ed line. The ele an in e nal mode is shown by ci cles joined by solid line 共 he i s cu e om below兲. FIG. 4. An ipe iodic BC: We show he di e ence be ween ␻ n pe iod⫺ ␻ n⫹1 pe iod s n(n⫽2,4,...,20). The s a s p ac ically a ze o o all n ep esen he in eg able sys em ( ⫽0), whe eas he long-dashed ( ⫽⫺0.02) and do ed ( ⫽0.02) lines co espond o nonin eg able cases 共nume ical esul s兲. Wi h iangles ( ⫽ ⫺0.02) and ci cles ( ⫽0.02) we plo he eigen equencies ob ained om he pe u ba ion heo y 共analy ical esul s兲. USING THE FINITE DOMAIN REMNANT OF THE... PHYSICAL REVIEW E 68, 036612 共2003兲 036612-5 symme ic a he bounda y, he ixed ones selec modes an- isymme ic a he bounda y, while he an ipe iodic BC allow o bo h 共c . Figs. 3 and 4兲. 共3兲An addi ional ea u e, equally impo an as 共1兲共espe- cially in iew o i s po en ial p edic i e powe 兲is he ac ha o he in eg able case o ⫽0, an ipe iodic BC essen- ially imply he p esence o double eigen alues. The di e - ence be ween he wo eigen alues is O(10⫺9) o all pai s 共excep o he cu o , disc e iza ion induced phenomena a he uppe end o he spec um which a e i ele an 兲. This is in sha p con as 共in pa icula , o small wa e numbe s兲, o e en mild b eakings o in eg abili y, as can be in e ed om Fig. 4. 共4兲S a emen s 共1兲and 共3兲abo e can be used in p edic i e o m and cons i u e he c i e ion 共algo i hm兲se o h in his wo k: o a gi en PDE/DDE model, we ind he s eady s a e cohe en s uc u e 共i.e., soli a y wa e兲in a ini e bu la ge domain. This can be done, e.g., by inding he exac solu ion o an ODE o nume ically pe o ming a New on- ype algo- i hm. Linea ize a ound he exac , ini e domain solu ion and s udy, in pa icula , he small wa e numbe s, close o he lowe edge o he spec um 共we assume ha he p oblem is monopa ame ic in wha ollows, bu i is clea ha he ap- plica ion o he c i e ion does no equi e ha 兲. I o a c i ical/singula alue o he pa ame e he ixed BC and ee BC 共small k) eigen alue spec a 共o he emnan o wha o he in ini e domain was he con inuous spec um兲essen ially coincide and he mul iplici y o an ipe iodic BC eigen alues becomes double, hen he model o his unique alue o he pa ame e can be ‘‘s ongly suspec ed’’ o be in eg able. We use he abo e exp ession, as we p o ide no igo ous p oo , bu only suppo ing 共bu a he uni e sal in dis inc models wi h dis inc ea u es/solu ions兲nume ical e idence o his s a emen . 共5兲We ha e also es ed he alidi y o hese esul s in Eq. 共2兲, in he icini y o he in eg able limi ⑀ ⫽0, wi h simila conclusions 关see Figs. 5 and 6兴. Indeed, in Fig. 5 we obse e he oscilla o y beha io o ␻ n ee⫺ ␻ n⫺1 ixed in he nonin eg able case, in Fig. 6, we show he case o pe iodic BC, whe e i can be clea ly seen ha i is only o he in eg able case ha he double eigen alue mul iplici y is ob ained. IV. CONCLUSIONS In conclusion, we ha e p oposed and used a es o e- ealing he po en ial in eg able na u e o a gi en model p ob- lem. By a ying he bounda y condi ions o a ini e domain compu a ion and examining he e ec s o such a ia ions in he 共con inuous- u ned-disc e e兲spec um, we ha e e ealed ha he small wa e numbe s ha e singula ways o espond- ing o he unique pa ame e alues o which he model is in eg able. These singula ea u es 关such as an app oxima e iden i ica ion o ixed wi h ee BC o small keigen alues and he double mul iplici y o eigen alues o pe iodic 共o an ipe iodic兲BC兴can be used o iden i y and single ou he in eg able beha io . We ha e p o ided wo model examples, espec i ely, o kinks and pulses and o a PDE and a DDE. Independen ly o he de ailed s uc u e o he model hese p ope ies ha e been iden i ied as uni e sal and ha e been suppo ed also by analy ical conside a ions. I would na u- ally be o in e es o explo e he po en ial use ulness o such a c i e ion in a ious mo e complex se ings. ACKNOWLEDGMENTS We would like o hank Jesu ´sSa ´nchez-Dehesa o he use ul discussion on he pe u ba ion heo y in he S u m- Liou ille p oblem. This wo k has been suppo ed by he Minis e io de Ciencia y Tecnologı ´a o Spain h ough G an No. BFM2001-3878-C02 and by he Jun a de Andalucı ´a un- de P ojec No. FQM-0207 共N.R.Q.兲. I has also been pa - ially suppo ed by NSF unde G an No. DMS-0204585, a Uni e si y o Massachuse s Facul y Resea ch G an and he Eppley Founda ion o Resea ch 共P.G.K.兲. FIG. 5. The oscilla o y beha io o ␻ n ee⫺ ␻ n⫺1 ixed in he nonin- eg able case ( ⑀ ⫽0.1) is shown o he i s wa e numbe s 共see he squa es joined by dashed line兲. The ci cles joined by solid line a e he esul s o he in eg able sys em ( ⑀ ⫽0). FIG. 6. Pe iodic BC o AL-DNLS 共Ablowi z-Ladik DNLS兲o Eq. 共2兲: The solid line a ze o ep esen s he di e ence be ween wo consecu i e equencies (n⫽2,4,...) o hein eg able AL la ice ( ⑀ ⫽0). The double mul iplici y o he equencies is des oyed as ⑀ is inc eased 共do ed, dashed, and do -dashed lines ep esen he non- in eg able cases o ⑀ ⫽0.1,0.5,1, espec i ely兲. P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-6 关1兴M.J. Ablowi z and H. Segu , Soli ons and he In e se Sca e - ing T ans o m 共SIAM, Philadelphia, 1981兲. 关2兴R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Mo is, Soli- ons and Nonlinea Wa e Equa ions 共Academic P ess, London, 1982兲. 关3兴A.C. Sco , Nonlinea Science 共Ox o d Uni e si y P ess, Ox- o d, 1999兲. 关4兴L.D. Faddee and L.A. Takh ajan, Hamil onian Me hods in he Theo y o Soli ons 共Sp inge -Ve lag, Be lin, 1987兲. 关5兴P. Painle e ´, C. R. Acad. Sci. 共Pa is兲130, 1112 共1900兲; A. Ra- mani, B. G amma icos, and A. Boun is, Phys. Rep. 180, 159 共1989兲; R. 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