Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation.

Gene alized a eling-wa e me hod, a ia ional app oach, and modi ied conse ed quan i ies o he pe u bed nonlinea Sch ödinge equa ion Niu ka R. Quin e o* Depa amen o de Física Aplicada I, E.U.P., Uni e sidad de Se illa, c/Vi gen de Á ica 7, 41011 Se illa, Spain F anz G. Me ens† Physikalisches Ins i u , Uni e si ä Bay eu h, D-95440 Bay eu h, Ge many A. R. Bishop Theo e ical Di ision and Cen e o Nonlinea S udies, Los Alamos Na ional Labo a o y, Los Alamos, New Mexico 87545, USA 共Recei ed 16 Ma ch 2010; e ised manusc ip ecei ed 8 June 2010; published 29 July 2010兲 The gene alized a eling wa e me hod 共GTWM兲is de eloped o he nonlinea Sch ödinge equa ion 共NLSE兲wi h gene al pe u ba ions in o de o ob ain he equa ions o mo ion o an a bi a y numbe o collec i e coo dina es. Rega dless o he pa icula ansa z ha is used, i is shown ha his al e na i e app oach is equi alen o he Lag angian o malism, bu has he ad an age ha only he Hamil onian o he unpe u bed sys em is equi ed, ins ead o he Lag angian o he pe u bed sys em. As an explici example, we ake 4 collec i e coo dina es, namely he posi ion, eloci y, ampli ude and phase o he soli on, and show ha he GTWM yields he same equa ions o mo ion as he pe u ba ion heo y based on he In e se Sca e ing T ans o m and as he ime a ia ion o he no m, i s momen o he no m, momen um, and ene gy o he pe u bed NLSE. DOI: 10.1103/PhysRe E.82.016606 PACS numbe 共s兲: 05.45.Y I. INTRODUCTION Wi h g ea gene ali y, and in many phenomena, such as sca e ing and di usion, soli ons, and o he nonlinea cohe - en exci a ions in ex ended sys ems beha e like pa icles 关1,2兴. This ac allows one o educe he in ini e numbe o deg ees o eedom o he ex ended sys ems o only a ew, when we a e in e es ed in he beha io o he soli on exci a- ions. E.g., he dynamics o opological soli ons o he non- linea Klein-Go don equa ions 共NLKGEs兲in he simples case can be desc ibed in e ms o one so-called collec i e coo dina e 共CC兲关3,4兴, ypically he cen e o he soli on. Mo e sophis ica ed Ansä ze ake in o accoun o he deg ees o eedom, using wo CCs 关5,6兴, h ee CCs 关7兴, o e en mo e 关8兴. The dynamics o non opological soli ons o he nonlinea Sch ödinge equa ion 共NLSE兲is mo e complica ed: The b igh one-soli on solu ion, which has an in e nal oscilla ion, depends on ou pa ame e s 关2兴. The e o e, ypically ou CCs, namely posi ion, ampli ude, eloci y and phase, ha e been used. The e a e se e al me hods o de e mine he equa ions o mo ion ha he CCs sa is y. Fo ins ance, o modi ied NLSEs se e al pe u ba ion heo ies can be applied 关9–12兴, o ce ain sys ems one can use he pe u bed in e se sca e - ing ans o m 共IST兲关13,14兴, and i a Lag angian densi y ex- is s one can de i e Lag ange equa ions, which a e he e olu- ion equa ions o he CCs 关15–17兴. Mo eo e , he ime a ia ion o Mconse ed quan i ies o he unpe u bed sys- em gene ally is ela ed wi h he e olu ion o MCCs 关18,19兴. The e is ano he me hod, he so-called gene alized a el- ing wa e me hod 共GTWM兲, in which a ce ain p ojec ion echnique is used o ob ain he equa ions o mo ion o he CCs. The GTWM was in oduced in a gene al way in Re . 关20兴: only he Hamil on equa ions o he unpe u bed sys em mus be known and he unpe u bed sys em need no be in- eg able. The me hod was applied o he ze o- empe a u e dynamics 关20兴and he he mal di usion 关21,22兴o magne ic o ices in he wo-dimensional 共2D兲aniso opic Heisenbe g model. The GTWM was also applied o he dynamics o opological soli ons in NLKGEs 关23兴. Toge he wi h he Rice ansa z 关5,6兴, which in oduces he posi ion and he wid h o he soli on as CCs, he me hod explained esonances due o he ac ion o an ac o ce 关23–25兴and desc ibed anspo phenomena o a che ype in sine-Go don and ␾ 4models wi h biha monic d i ing and damping 关26,27兴. The sine- Go don model was applied, e.g., o long Josephson junc ions, whe e he luxons play he ole o he soli ons, and he a che e ec was con i med by expe imen s 关28兴. The aim o his wo k is o de elop he GTWM o an a bi a y ansa z wi h MCCs o he pe u bed NLSE, see Sec. II. Fu he mo e, we show in Sec. III ha in cases whe e he Lag angian densi y o he pe u bed sys em exis s, he GTWM is equi alen o he a ia ional app oach. In pa icu- la , by using only 4 CCs we show in Sec. IV ha he GTWM is also equi alen o he esul s ob ained om he pe u ba- ion heo y based on he in e se sca e ing ans o m 关14兴, om he adiaba ic pe u ba ion heo y 关29兴and om he a ia ion o he no m, he i s momen o he no m, he momen um, and he ene gy o he pe u bed NLSE. Finally in Sec. V, we summa ize ou main esul s. *[email p o ec ed] †[email p o ec ed] PHYSICAL REVIEW E 82, 016606 共2010兲 1539-3755/2010/82共1兲/016606共6兲©2010 The Ame ican Physical Socie y016606-1 II. GENERALIZED TRAVELING WAVE METHOD The pe u bed NLSE ac ually consis s o wo equa ions o he eal and imagina y pa s, o o u共x, 兲and uⴱ共x, 兲. Namely, iu +uxx +2兩u兩2u=R关u共x, 兲;x, 兴,共1兲 and i s complex conjuga e equa ion, whe e R关u共x, 兲;x, 兴 ep esen s a gene al pe u ba ion, which may also depend on uⴱand he spa ial de i a i es o uand uⴱ. This sys em has e y many applica ions in p ac ically all ields o physics, which a e lis ed and discussed in se e al e iews a icles 关14–16,30兴. In he case o op ical soli ons, and xa e e- placed by he p opaga ion dis ance and he pulse du a ion, espec i ely. Fo ou pu pose, we ew i e Eq. 共1兲 iu = ␦ H0 ␦ uⴱ+R关u共x, 兲;x, 兴,共2兲 whe e H0= 冕 −⬁ +⬁ dxH0= 冕 −⬁ +⬁ dx共uxux ⴱ−u2uⴱ2兲.共3兲 Fo he ollowing, only his Hamil onian o he unpe u bed sys em mus be known. We now assume ha he ime depen- dence o u共x, 兲and uⴱ共x, 兲in Eq. 共2兲and i s complex con- juga e equa ion only appea s ia a se o M eal collec i e coo dina es 兵Y1共 兲,Y2共 兲,...YM共 兲其ªY ជ 共 兲 u共x, 兲=u关x,Y ជ 共 兲兴,uⴱ共x, 兲=uⴱ关x,Y ជ 共 兲兴.共4兲 Then, we mul iply Eq. 共2兲by ⳵ uⴱ/ ⳵ Yn, and i s complex con- juga e equa ion by ⳵ u/ ⳵ Yn, add he esul ing equa ions and in eg a e o e he sys em, yielding 兺 j=1 M IYnYjY ˙j=Fn共Y ជ 兲+Rn共Y ជ 兲,n=1,2, ...M,共5兲 wi h IYnYj= 冕 −⬁ +⬁ dx i 冋 ⳵ u ⳵ Yn ⳵ uⴱ ⳵ Yj − ⳵ uⴱ ⳵ Yn ⳵ u ⳵ Yj 册 ,共6兲 Fn共Y ជ 兲=− 冕 −⬁ +⬁ dx 冋 ␦ H0 ␦ uⴱ ⳵ uⴱ ⳵ Yn + ␦ H0 ␦ u ⳵ u ⳵ Yn 册 =− 冕 −⬁ +⬁ dx ⳵ H0 ⳵ Yn =− ⳵ ⳵ Yn H0,共7兲 Rn共Y ជ 兲=− 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ Yn +Rⴱ ⳵ u ⳵ Yn 冊 ,共8兲 whe e he o e do deno es he de i a i e wi h espec o ime. Equa ions 共5兲–共8兲 ep esen a se o M i s -o de ODEs o ou MCCs. We no e ha o ce ain sys ems i is neces- sa y o assume ha he ime dependence o he ields can also appea ia Y ˙1,Y ˙2,...,Y ˙M, which a e o mally ea ed in he same way as he CCs. This yields Msecond-o de ODEs o he MCCs. Examples a e magne ic o ices 关20兴and opo- logical soli ons in NLKGEs 关23,31兴; in bo h cases he wid h o he exci a ions depends on he eloci y o he exci a ions. In o de o ha e an example which can be es ed by he pe u ba ion heo y based on he IST 关14兴we now ake he one-soli on solu ion o he unpe u bed NLSE and make he ollowing speci ic ansa z o u共x,Y ជ 共 兲兲: u共x, 兲=2i ␩ sech关2 ␩ 共x− ␨ 兲兴e−i关2 ␰ 共x− ␨ 兲+⌽兴,共9兲 wi h he 4 CCs soli on posi ion ␨ 共 兲, phase ⌽共 兲, ampli ude ␩ 共 兲, and eloci y ␰ 共 兲. The soli on ene gy eads H0=16 3 ␩ 共3 ␰ 2− ␩ 2兲.共10兲 Se ing Y1= ␨ ,Y2=⌽,Y3= ␩ and Y4= ␰ in Eqs. 共5兲–共8兲,we ob ain o n=1 8 ␩ ˙ ␰ +8 ␩ ␰ ˙= 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵␨ +Rⴱ ⳵ u ⳵␨ 冊 ,共11兲 whe e 2 ␩ is he soli on mass, and −4 ␰ 共 兲 he soli on eloci y 关30兴. The e o e his equa ion has he o m o he New onian equa ion o mo ion o a poin pa icle wi h a iable mass. The hs o Eq. 共11兲 ep esen s he ime-dependen o ce ob ained by means o he a e age o he pe u ba ion Ro e space, whe e he “weigh unc ion” is jus he a ia ion o he ansa z wi h espec o he i s CC, he soli on posi ion. In addi ion o he poin -pa icle ep esen a ion, he e olu ion o ␨ and ⌽ akes in o accoun o he deg ees o eedom. Then, in a simila way, i.e., se ing Y1= ␨ ,Y2=⌽,Y3= ␩ and Y4= ␰ in Eqs. 共5兲–共8兲, we ob ain o n=2,3,4 4 ␩ ˙=− 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ ␾ +Rⴱ ⳵ u ⳵ ␾ 冊 ,共12兲 4共⌽ ˙−2 ␰␨ ˙兲⬅4共 ␾ ˙+2 ␨␰ ˙兲 =−16共 ␩ 2− ␰ 2兲+ 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ ␩ +Rⴱ ⳵ u ⳵ ␩ 冊 , 共13兲 8 ␩ ␨ ˙=−32 ␩ ␰ − 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵␰ +Rⴱ ⳵ u ⳵␰ 冊 ,共14兲 espec i ely, whe e ⌽= ␾ +2 ␨␰ . Hence, he e olu ion o ␩ and ␨ is gi en di ec ly by Eqs. 共12兲and 共14兲, espec i ely. Subs i u ing ␩ ˙ om Eq. 共12兲in Eq. 共11兲 he equa ion o mo ion o ␰ is a ained. Finally, using he ob ained equa ion o ␰ ˙in Eq. 共13兲 he ime e olu ion o ⌽is ound. Equa ions 共11兲–共14兲a e equi alen o he esul s o he adiaba ic pe - u ba ion heo y 关29兴. They a e also equi alen o he Eqs. 共3.17兲–共3.20兲o Re . 关14兴 o ␩ ˙, ␰ ˙, ␨ ˙, and ␾ ˙ob ained by he IST wi h a bi a y pe u ba ions. Howe e , he GTWM has he ollowing ad an ages: 共1兲I does no equi e ha he IST can be applied o he unpe u bed sys em. In ac , he speci ic o m o H0in Eq. 共3兲, which is ela ed o he unpe u bed NLSE, is no used in he de i a ion o he e olu ion Eqs. 共5兲 o he CCs. QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲 016606-2 共2兲The GTWM wo ks o an a bi a y numbe o CCs. Fo example he ansa z o Re . 关37兴wi h 6 CCs can be used. See also he ema ks a he end o he conclusions sec ion. 共3兲The p ocedu e which yields he e olu ion Eqs. 共5兲 o he CCs is ex emely concise. A e ha ing chosen a speci ic ansa z, he calcula ion o he in eg als Eq. 共6兲and he o ces Eq. 共7兲is s aigh o wa d. The same holds o he pe u ba- ion e ms Eq. 共8兲when he pe u ba ion Rhas been speci- ied. III. VARIATIONAL APPROACH AND GTWM An ad an age o he GTWM is ela ed o he ac ha i can be applied in cases whe e he Lag angian densi y L共u,uⴱ,ux,ux ⴱ,u ,u ⴱ兲o he pe u bed sys em is unknown. Howe e , i Lexis s and i , in addi ion, Land he ansa z Eqs. 共4兲sa is y ce ain condi ions 关see Eqs. 共22兲and 共23兲 below兴, we show ha he a ia ional app oach and he GTWM a e equi alen , i.e., bo h me hods yield he same equa ions o mo ion o he CCs. In o de o p o e his s a emen , we sepa a e he pe u ba- ion R关u共x, 兲;x, 兴in wo pa s R关u共x, 兲;x, 兴=−i ␤ u+B关u共x, 兲;x, 兴,共15兲 such ha he dissipa ion in he sys em appea s only in he i s e m, −i ␤ u. We assume ha he pe u bed NLSE Eq. 共1兲 is equi alen o an Eule -Lag ange equa ion, gene alized by a dissipa i e e m on he hs, d d ⳵ L ⳵ u ⴱ+d dx ⳵ L ⳵ ux ⴱ− ⳵ L ⳵ uⴱ= ␦ F ␦ u ⴱ,共16兲 wi h he dissipa ion unc ion F=−i ␤ 共uu ⴱ−uⴱu 兲,共17兲 and simila ly o he complex conjuga e o Eq. 共1兲. We no e ha mo e complica ed dissipa ion e ms han he simple e m −i ␤ uin Eq. 共15兲can also be ea ed by gene alizing he Eule -Lag ange o malism 关17兴. Now, inse ing he ansa z Eq. 共4兲in Land F关32,33兴and in eg a ing, we ob ain L= 冕 −⬁ +⬁ dxL共u,uⴱ,ux,ux ⴱ,u ,u ⴱ兲=L共Y ជ ;Y ជ ˙兲,共18兲 F= 冕 −⬁ +⬁ dxF共u,uⴱ,ux,ux ⴱ,u ,u ⴱ兲=F共Y ជ ;Y ជ ˙兲.共19兲 Hence, he Mcollec i e coo dina es sa is y Mgene alized Lag ange equa ions d d ⳵ L ⳵ Y ˙n − ⳵ L ⳵ Yn − ⳵ F ⳵ Y ˙n =0, 共20兲 o equi alen ly 冕 −⬁ +⬁ dx 冋 d d ⳵ L ⳵ Y ˙n − ⳵ L ⳵ Yn − ⳵ F ⳵ Y ˙n 册 =0, 共21兲 whe e n=1,...,M. F om now on we deno e Y=Ynand as- sume 兵see Eq. 共4兲in 关32兴其 ha 冋 ⳵ L ⳵ ux ⳵ u ⳵ Y 册 x→+⬁ − 冋 ⳵ L ⳵ ux ⳵ u ⳵ Y 册 x→−⬁ =0, 共22兲 冋 ⳵ L ⳵ ux ⴱ ⳵ uⴱ ⳵ Y 册 x→+⬁ − 冋 ⳵ L ⳵ ux ⴱ ⳵ uⴱ ⳵ Y 册 x→−⬁ =0. 共23兲 As he ollowing ela ions hold d d ⳵ L ⳵ Y ˙=d d 冉 ⳵ L ⳵ u ⳵ u ⳵ Y 冊 +d d 冉 ⳵ L ⳵ u ⴱ ⳵ uⴱ ⳵ Y 冊 ,共24兲 ⳵ L ⳵ Y= ⳵ L ⳵ u ⳵ u ⳵ Y+ ⳵ L ⳵ uⴱ ⳵ uⴱ ⳵ Y+ ⳵ L ⳵ ux ⳵ ux ⳵ Y + ⳵ L ⳵ ux ⴱ ⳵ ux ⴱ ⳵ Y+ ⳵ L ⳵ u ⳵ u ⳵ Y+ ⳵ L ⳵ u ⴱ ⳵ u ⴱ ⳵ Y,共25兲 ⳵ F ⳵ Y ˙= ⳵ F ⳵ u ⳵ u ⳵ Y+ ⳵ F ⳵ u ⴱ ⳵ uⴱ ⳵ Y,共26兲 and aking in o accoun he ela ions Eqs. 共22兲and 共23兲, Eq. 共21兲becomes 冕 −⬁ +⬁ dx 再 d d 冉 ⳵ L ⳵ u 冊 +d dx 冉 ⳵ L ⳵ ux 冊 − ⳵ L ⳵ u− ⳵ F ⳵ u 冎 ⳵ u ⳵ Y + 冕 −⬁ +⬁ dx 再 d d 冉 ⳵ L ⳵ u ⴱ 冊 +d dx 冉 ⳵ L ⳵ ux ⴱ 冊 − ⳵ L ⳵ uⴱ− ⳵ F ⳵ u ⴱ 冎 ⳵ uⴱ ⳵ Y=0. 共27兲 Inse ing he Lag angian densi y Land he dissipa ion unc- ion Fin Eq. 共27兲, he i s and second cu ly b acke s become he NLS Eq. 共1兲and i s co esponding complex conjuga e equa ion, espec i ely 共up o a cons an ac o 兲, whe e Ris gi en by Eq. 共15兲. Then, Eq. 共27兲 ep esen s he in eg a ion o he sum o he i s NLSE Eq. 共1兲mul iplied by ⳵ u/ ⳵ Yand he second NLSE, i.e., he complex conjuga e o Eq. 共1兲, mul iplied by ⳵ uⴱ/ ⳵ Y. This p ojec ion p ocedu e is exac ly he same as ha which has p oduced he MEqs. 共5兲in Sec. II. Thus, he Lag angian app oach is equi alen o he GTWM and ice e sa. Howe e , in p ac ice he GTWM equi es mo e wo k i M⬎3, because M共M−1兲/2 in eg als IYnYjand Min eg als Rn共Y ជ 兲in Eqs. 共6兲and 共8兲, espec i ely, mus be calcula ed. In he Lag angian me hod one has o pe o m only M+3 in eg als o ob ain Land F. On he o he hand, he GTWM is mo e gene al since he Lag angian den- si y o he pe u bed sys em need no be known. IV. GTWM AND MODIFIED CONSERVED QUANTITIES A e y peculia p ope y o GTWMs is ela ed o i s ela ionship wi h he so-called modi ied conse a ion laws 共MCL兲共 he ime e olu ion o he quan i ies which a e con- se ed o he unpe u bed sys em兲关18兴. Such an equi alence was al eady shown o NLKGEs in 关23兴since usually he GENERALIZED TRAVELING-WAVE METHOD,…PHYSICAL REVIEW E 82, 016606 共2010兲 016606-3 equa ions o mo ion o M CCs a e ela ed wi h he i s M MCLs. In his sec ion, we show ha in he case o he NLSE he GTWM wi h he ansa z u共x, 兲=u共x− ␨ ,⌽, ␩ , ␰ 兲,uⴱ共x, 兲=uⴱ共x− ␨ ,⌽, ␩ , ␰ 兲, 共28兲 yields a i s only 3 MCLs, namely he ime a ia ion o he no m, he momen um, and he ene gy. In Eq. 共28兲 ␨ deno es he soli on posi ion and ⌽ ep esen s a phase, 共so ha ⳵ u/ ⳵ ⌽=−iu and ⳵ uⴱ/ ⳵ ⌽=iuⴱ兲. We s ess ha we need no ye make a speci ic ansa z o u. Wi h M=4 and se ing n =1 in Eqs. 共5兲–共8兲we ob ain 兺 j=1 4 I ␨ YjY ˙j=dP d + 冕 −⬁ +⬁ dx ⳵ ⳵ x i 2关u uⴱ−uu ⴱ兴,共29兲 F1共 ␨ ,⌽, ␩ , ␰ 兲= 冕 −⬁ +⬁ dx ⳵ H0 ⳵ x,共30兲 R1共 ␨ ,⌽, ␩ , ␰ 兲= 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ x+Rⴱ ⳵ u ⳵ x 冊 ,共31兲 whe e Pis he momen um P= 冕 −⬁ +⬁ dx i 2关uux ⴱ−uⴱux兴.共32兲 Subs i u ing Eqs. 共29兲–共31兲in Eq. 共5兲and aking in o accoun ha 冕 −⬁ +⬁ dx ⳵ ⳵ x 再 i 2关u uⴱ−uu ⴱ兴−H0 冎 =0, 共33兲 we ob ain dP d = 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ x+Rⴱ ⳵ u ⳵ x 冊 ,共34兲 which ep esen s he a ia ion o he momen um wi h espec o ime o he pe u bed NLSE Eq. 共1兲and i s complex conjuga e equa ion. Taking n=2 in Eqs. 共5兲–共8兲, we ob ain 兺 j=1 4 I⌽YjY ˙j=−dN d ,N= 冕 −⬁ +⬁ dx兩u兩2,共35兲 F2共 ␨ ,⌽, ␩ , ␰ 兲=− ⳵ H0 ⳵ ⌽=− ⳵ ⳵ ⌽ 冕 −⬁ +⬁ dx共兩ux兩2−兩u兩4兲=0, 共36兲 R2共 ␨ ,⌽, ␩ , ␰ 兲=− 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ ⌽+Rⴱ ⳵ u ⳵ ⌽ 冊 =− 冕 −⬁ +⬁ dx i共Rⴱu−Ruⴱ兲.共37兲 Hence, subs i u ing Eqs. 共35兲–共37兲in Eq. 共5兲ou second modi ied conse a ion law eads dN d = 冕 −⬁ +⬁ dx i共Rⴱu−Ruⴱ兲,共38兲 and ep esen s he a ia ion o he no m N. Finally, we inse in Eqs. 共5兲–共8兲Y1= ␨ ,Y2=⌽,Y3= ␩ , and Y4= ␰ , wi h M=4. Then, we mul iply hese equa ion by ␨ ˙,⌽ ˙, ␩ ˙, and ␰ ˙, espec i ely; adding hem yields 兺 n=1 4 兺 j=1 4 IYnYjY ˙jY ˙n=0, 共39兲 F1 ␨ ˙+F2⌽ ˙+F3 ␩ ˙+F4 ␰ ˙=−dH0 d ,共40兲 R1 ␨ ˙+R2⌽ ˙+R3 ␩ ˙+R4 ␰ ˙=− 冕 −⬁ +⬁ dx共Ru ⴱ+Rⴱu 兲,共41兲 so ha he e olu ion o he ene gy is gi en by dH0 d =− 冕 −⬁ +⬁ dx共Ru ⴱ+Rⴱu 兲,共42兲 which can be w i en as dH0 d =−兺 j=1 4 Y ˙j 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ Yj +Rⴱ ⳵ u ⳵ Yj 冊 .共43兲 Equa ion 共43兲 ep esen s he hi d MCL o NLSE. We would like o ind a ou h MCL, since we a e using ou CC equa ions om ou GTWM. I is na u al o expec ha hese equa ions a e ela ed no only o he i s h ee conse a ion laws bu also o he ou h one. The in ini e sequence o conse a ion laws o he unpe u bed NLSE can be ob ained ia he ollowing ecu ence ela ion o densi- ies, which was ob ained in he amewo k o he IST 关34兴, bm+1 =uⴱd dx 冉 bm uⴱ 冊 +兺 k+j=m−1 bkbj,共44兲 wi h b0=兩u兩2,m=0,1,....Asimila ela ion, wi h bm+1 ⴱ, holds o he complex conjuga e o Eq. 共1兲wi h R=0. In his way, o m=0 he momen um densi y P=−共i/2兲共b1−b1 ⴱ兲is ob- ained 关34兴, which is iden ical wi h ha in Eq. 共32兲. Fo m =1, we ob ain he ene gy densi y and o m=2 a new densi y, ␳ 1=−共i/2兲共b3−b3 ⴱ兲=i 2兵3兩u兩2共uux ⴱ−uxuⴱ兲+ux ⴱuxx −uxuxx ⴱ其. 共45兲 Using he one-soli on solu ion o he unpe u bed NLSE, i.e., Equa ion 共9兲wi h cons an eloci y and ampli ude and QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲 016606-4 he linea unc ions in ime ␨ = ␨ 0−4 ␰ and ⌽=⌽0 −4共 ␰ 2+ ␩ 2兲 , his yields a new conse ed quan i y H1 =兰−⬁ +⬁dx ␳ 1共x, 兲=32 ␩ ␰ 共 ␰ 2− ␩ 2兲. Due o he pe u ba ions R and Rⴱin Eq. 共1兲and i s complex conjuga e equa ion, espec- i ely, he e olu ion o H1is de e mined by 关18兴 dH1 d = 冕 −⬁ +⬁ dx关Rⴱ共6兩u兩2ux+uxxx兲+R共6兩u兩2ux ⴱ+uxxx ⴱ兲兴, 共46兲 whe e H1is he Hamil onian o he i s in eg able hie a chy o he NLSE 关35兴. Compa ing he hs o Eq. 共8兲wi h he hs o Eq. 共46兲we ealize ha in he o me equa ion he pe u - ba ion Ris mul iplied by he i s de i a i e o uⴱwi h e- spec o a gi en CC, whe eas in he la e one he hi d spa ial de i a i e o uⴱappea s. So, he MCL Eq. 共46兲is no equi a- len o any equa ion o combina ion o equa ions o he sys- em Eqs. 共5兲–共8兲ob ained by means o GTWM. Howe e , we can achie e ou goal by conside ing he e olu ion o he i s momen o he no m 关19兴, dN1 d =2P+ 冕 −⬁ +⬁ dx x共Rⴱu−Ruⴱ兲,N1= 冕 −⬁ +⬁ dx x兩u兩2. 共47兲 We now show ha Eq. 共47兲 oge he wi h he i s h ee MCLs yields he ou CC Eqs. 共11兲–共14兲. He e we use he speci ic ansa z Eq. 共9兲 o which he no m N=4 ␩ ,共48兲 he i s momen o he no m, N1=4 ␩ ␨ ,共49兲 he momen um P=−8 ␩ ␰ ,共50兲 and he ene gy is gi en by Eq. 共10兲. Subs i u ing Eqs. 共48兲–共50兲and 共10兲in he Eqs. 共38兲,共34兲,共47兲, and 共43兲, espec i ely, we ob ain 4 ␩ ˙= 冕 −⬁ +⬁ dx i共Rⴱu−Ruⴱ兲,共51兲 −8 ␩ ␰ ˙−8 ␰ ␩ ˙= 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ x+Rⴱ ⳵ u ⳵ x 冊 ,共52兲 4 ␩ ␨ ˙+4 ␨ ␩ ˙=−16 ␩ ␰ +i 冕 −⬁ +⬁ dx x共Rⴱu−Ruⴱ兲,共53兲 −16共 ␩ 2− ␰ 2兲 ␩ ˙+32 ␩ ␰␰ ˙=−兺 j=1 4 Y ˙j 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ Yj +Rⴱ ⳵ u ⳵ Yj 冊 . 共54兲 F om Eq. 共51兲and he iden i y ⳵ u/ ⳵␰ =−2i共x− ␨ 兲u, Eq. 共53兲 becomes 8 ␩ ␨ ˙=−32 ␩ ␰ − 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵␰ +Rⴱ ⳵ u ⳵␰ 冊 ,共55兲 which p o ides he e olu ion equa ion o ␨ ˙. Taking in o ac- coun Eqs. 共51兲,共52兲, and 共55兲, Eq. 共54兲becomes ␩ ˙ 再 4共⌽ ˙−2 ␰␨ ˙兲−16共 ␰ 2− ␩ 2兲− 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ ␩ +Rⴱ ⳵ u ⳵ ␩ 冊冎 =0, 共56兲 yielding he e olu ion equa ion o ⌽since ␩ e ol es in ime. F om Eqs. 共51兲and 共52兲we ob ain he equa ion o mo ion o ␩ and ␰ . F om Eqs. 共55兲and 共56兲, he equa ions o ␨ and ⌽can be ob ained. In ac , hese ou equa ions ag ee wi h Eqs. 共11兲–共14兲ob ained ia GTWM. V. CONCLUSIONS We ha e de eloped he GTWM o he pe u bed NLS Eqs. 共1兲and he complex conjuga e o Eq. 共1兲共 his is di e - en om ea lie applica ions o he me hod which s a ed om he Hamil on equa ions o magne ic o ices and o soli ons in NLKGEs兲. The GTWM is based on a p ojec ion echnique: The NLSEs Eq. 共2兲and he complex conjuga e o Eq. 共2兲a e mul iplied by de i a i es o uⴱand uwi h espec o one o he CCs, espec i ely. Then he esul ing equa ions a e added and in eg a ed o e he sys em. This yields he e olu ion equa ions o an a bi a y numbe o CCs, Eqs. 共5兲–共8兲, wi hou using a speci ic ansa z. This me hod is mo e gene al han he a ia ional app oach because i is no nec- essa y o know he Lag angian densi y Lo he pe u bed sys em. E.g., see Eq. 共1兲o 关36兴, whe e he Lag angian den- si y does no exis , bu he Hamil onian H0o he unpe - u bed sys em is known. In cases whe e Lexis s and condi- ions Eqs. 共22兲and 共23兲a e sa is ied, we ha e shown ha he GTWM and he a ia ional o malism a e equi alen , i.e., om bo h heo ies he same e olu ion equa ions o he CCs a e ob ained. Using 4 collec i e coo dina es wi h he ansa z Eq. 共9兲, namely, he posi ion ␨ , eloci y ␰ , ampli ude ␩ and phase ⌽ o he soli on, we ha e explici ly shown ha he GTWM is equi alen o he esul s ob ained om he IST, om he adiaba ic pe u ba ion heo y and om he ime a ia ion o he no m, he i s momen o he no m, he momen um, and he ene gy o he sys em. Howe e , we s ess ha he GTWM is e y concise, and mo e gene al han he IST and he ime- a ia ion o he conse ed quan i ies since i can be applied when we ha e an ansa z wi h mo e han 4 CCs 共see he ansä ze wi h 5 CCs in Re s. 关36,38兴 o soli on p opaga- ion in op ical ibe s and wi h 6 CCs used in 关37兴兲. 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