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

Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation Niurka R. Quintero* Departamento de Física Aplicada I, E.U.P., Universidad de Sevilla, c/Virgen de África 7, 41011 Sevilla, Spain Franz G. Mertens† Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany A. R. Bishop Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 共Received 16 March 2010; revised manuscript received 8 June 2010; published 29 July 2010兲 The generalized traveling wave method 共GTWM兲is developed for the nonlinear Schrödinger equation 共NLSE兲with general perturbations in order to obtain the equations of motion for an arbitrary number of collective coordinates. Regardless of the particular ansatz that is used, it is shown that this alternative approach is equivalent to the Lagrangian formalism, but has the advantage that only the Hamiltonian of the unperturbed system is required, instead of the Lagrangian for the perturbed system. As an explicit example, we take 4 collective coordinates, namely the position, velocity, amplitude and phase of the soliton, and show that the GTWM yields the same equations of motion as the perturbation theory based on the Inverse Scattering Transform and as the time variation of the norm, first moment of the norm, momentum, and energy for the perturbed NLSE. DOI: 10.1103/PhysRevE.82.016606 PACS number共s兲: 05.45.Yv I. INTRODUCTION With great generality, and in many phenomena, such as scattering and diffusion, solitons, and other nonlinear coherent excitations in extended systems behave like particles 关1,2兴. This fact allows one to reduce the infinite number of degrees of freedom of the extended systems to only a few, when we are interested in the behavior of the soliton excitations. E.g., the dynamics of topological solitons of the nonlinear Klein-Gordon equations 共NLKGEs兲in the simplest case can be described in terms of one so-called collective coordinate 共CC兲关3,4兴, typically the center of the soliton. More sophisticated Ansätze take into account other degrees of freedom, using two CCs 关5,6兴, three CCs 关7兴, or even more 关8兴. The dynamics of nontopological solitons of the nonlinear Schrödinger equation 共NLSE兲is more complicated: The bright one-soliton solution, which has an internal oscillation, depends on four parameters 关2兴. Therefore, typically four CCs, namely position, amplitude, velocity and phase, have been used. There are several methods to determine the equations of motion that the CCs satisfy. For instance, for modified NLSEs several perturbation theories can be applied 关9–12兴, for certain systems one can use the perturbed inverse scattering transform 共IST兲关13,14兴, and if a Lagrangian density exists one can derive Lagrange equations, which are the evolution equations of the CCs 关15–17兴. Moreover, the time variation of Mconserved quantities of the unperturbed system generally is related with the evolution of MCCs 关18,19兴. There is another method, the so-called generalized traveling wave method 共GTWM兲, in which a certain projection technique is used to obtain the equations of motion of the CCs. The GTWM was introduced in a general way in Ref. 关20兴: only the Hamilton equations of the unperturbed system must be known and the unperturbed system need not be integrable. The method was applied to the zero-temperature dynamics 关20兴and the thermal diffusion 关21,22兴of magnetic vortices in the two-dimensional 共2D兲anisotropic Heisenberg model. The GTWM was also applied to the dynamics of topological solitons in NLKGEs 关23兴. Together with the Rice ansatz 关5,6兴, which introduces the position and the width of the soliton as CCs, the method explained resonances due to the action of an ac force 关23–25兴and described transport phenomena of ratchet type in sine-Gordon and ␾ 4models with biharmonic driving and damping 关26,27兴. The sine- Gordon model was applied, e.g., to long Josephson junctions, where the fluxons play the role of the solitons, and the ratchet effect was confirmed by experiments 关28兴. The aim of this work is to develop the GTWM for an arbitrary ansatz with MCCs for the perturbed NLSE, see Sec. II. Furthermore, we show in Sec. III that in cases where the Lagrangian density for the perturbed system exists, the GTWM is equivalent to the variational approach. In particular, by using only 4 CCs we show in Sec. IV that the GTWM is also equivalent to the results obtained from the perturbation theory based on the inverse scattering transform 关14兴, from the adiabatic perturbation theory 关29兴and from the variation of the norm, the first moment of the norm, the momentum, and the energy of the perturbed NLSE. Finally in Sec. V, we summarize our main results. *[email protected] †[email protected] PHYSICAL REVIEW E 82, 016606 共2010兲 1539-3755/2010/82共1兲/016606共6兲©2010 The American Physical Society016606-1 II. GENERALIZED TRAVELING WAVE METHOD The perturbed NLSE actually consists of two equations for the real and imaginary parts, or for u共x,t兲and uⴱ共x,t兲. Namely, iut+uxx +2兩u兩2u=R关u共x,t兲;x,t兴,共1兲 and its complex conjugate equation, where R关u共x,t兲;x,t兴 represents a general perturbation, which may also depend on uⴱand the spatial derivatives of uand uⴱ. This system has very many applications in practically all fields of physics, which are listed and discussed in several reviews articles 关14–16,30兴. In the case of optical solitons, tand xare replaced by the propagation distance and the pulse duration, respectively. For our purpose, we rewrite Eq. 共1兲 iut= ␦ H0 ␦ uⴱ+R关u共x,t兲;x,t兴,共2兲 where H0= 冕 −⬁ +⬁ dxH0= 冕 −⬁ +⬁ dx共uxux ⴱ−u2uⴱ2兲.共3兲 For the following, only this Hamiltonian of the unperturbed system must be known. We now assume that the time dependence of u共x,t兲and uⴱ共x,t兲in Eq. 共2兲and its complex conjugate equation only appears via a set of Mreal collective coordinates 兵Y1共t兲,Y2共t兲,...YM共t兲其ªY ជ 共t兲 u共x,t兲=u关x,Y ជ 共t兲兴,uⴱ共x,t兲=uⴱ关x,Y ជ 共t兲兴.共4兲 Then, we multiply Eq. 共2兲by ⳵ uⴱ/ ⳵ Yn, and its complex conjugate equation by ⳵ u/ ⳵ Yn, add the resulting equations and integrate over the system, yielding 兺 j=1 M IYnYjY ˙j=Fn共Y ជ 兲+Rn共Y ជ 兲,n=1,2, ...M,共5兲 with 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 where the overdot denotes the derivative with respect to time. Equations 共5兲–共8兲represent a set of Mfirst-order ODEs for our MCCs. We note that for certain systems it is necessary to assume that the time dependence of the fields can also appear via Y ˙1,Y ˙2,...,Y ˙M, which are formally treated in the same way as the CCs. This yields Msecond-order ODEs for the MCCs. Examples are magnetic vortices 关20兴and topological solitons in NLKGEs 关23,31兴; in both cases the width of the excitations depends on the velocity of the excitations. In order to have an example which can be tested by the perturbation theory based on the IST 关14兴we now take the one-soliton solution of the unperturbed NLSE and make the following specific ansatz for u共x,Y ជ 共t兲兲: u共x,t兲=2i ␩ sech关2 ␩ 共x− ␨ 兲兴e−i关2 ␰ 共x− ␨ 兲+⌽兴,共9兲 with the 4 CCs soliton position ␨ 共t兲, phase ⌽共t兲, amplitude ␩ 共t兲, and velocity ␰ 共t兲. The soliton energy reads H0=16 3 ␩ 共3 ␰ 2− ␩ 2兲.共10兲 Setting Y1= ␨ ,Y2=⌽,Y3= ␩ and Y4= ␰ in Eqs. 共5兲–共8兲,we obtain for n=1 8 ␩ ˙ ␰ +8 ␩ ␰ ˙= 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵␨ +Rⴱ ⳵ u ⳵␨ 冊 ,共11兲 where 2 ␩ is the soliton mass, and −4 ␰ 共t兲the soliton velocity 关30兴. Therefore this equation has the form of the Newtonian equation of motion for a point particle with variable mass. The rhs of Eq. 共11兲represents the time-dependent force obtained by means of the average of the perturbation Rover space, where the “weight function” is just the variation of the ansatz with respect to the first CC, the soliton position. In addition to the point-particle representation, the evolution of ␨ and ⌽takes into account other degrees of freedom. Then, in a similar way, i.e., setting Y1= ␨ ,Y2=⌽,Y3= ␩ and Y4= ␰ in Eqs. 共5兲–共8兲, we obtain for 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 respectively, where ⌽= ␾ +2 ␨␰ . Hence, the evolution of ␩ and ␨ is given directly by Eqs. 共12兲and 共14兲, respectively. Substituting ␩ ˙from Eq. 共12兲in Eq. 共11兲the equation of motion for ␰ is attained. Finally, using the obtained equation for ␰ ˙in Eq. 共13兲the time evolution of ⌽is found. Equations 共11兲–共14兲are equivalent to the results of the adiabatic perturbation theory 关29兴. They are also equivalent to the Eqs. 共3.17兲–共3.20兲of Ref. 关14兴for ␩ ˙, ␰ ˙, ␨ ˙, and ␾ ˙obtained by the IST with arbitrary perturbations. However, the GTWM has the following advantages: 共1兲It does not require that the IST can be applied to the unperturbed system. In fact, the specific form of H0in Eq. 共3兲, which is related to the unperturbed NLSE, is not used in the derivation of the evolution Eqs. 共5兲for the CCs. QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲 016606-2 共2兲The GTWM works for an arbitrary number of CCs. For example the ansatz of Ref. 关37兴with 6 CCs can be used. See also the remarks at the end of the conclusions section. 共3兲The procedure which yields the evolution Eqs. 共5兲for the CCs is extremely concise. After having chosen a specific ansatz, the calculation of the integrals Eq. 共6兲and the forces Eq. 共7兲is straightforward. The same holds for the perturbation terms Eq. 共8兲when the perturbation Rhas been specified. III. VARIATIONAL APPROACH AND GTWM An advantage of the GTWM is related to the fact that it can be applied in cases where the Lagrangian density L共u,uⴱ,ux,ux ⴱ,ut,ut ⴱ兲of the perturbed system is unknown. However, if Lexists and if, in addition, Land the ansatz Eqs. 共4兲satisfy certain conditions 关see Eqs. 共22兲and 共23兲 below兴, we show that the variational approach and the GTWM are equivalent, i.e., both methods yield the same equations of motion for the CCs. In order to prove this statement, we separate the perturbation R关u共x,t兲;x,t兴in two parts R关u共x,t兲;x,t兴=−i ␤ u+B关u共x,t兲;x,t兴,共15兲 such that the dissipation in the system appears only in the first term, −i ␤ u. We assume that the perturbed NLSE Eq. 共1兲 is equivalent to an Euler-Lagrange equation, generalized by a dissipative term on the rhs, d dt ⳵ L ⳵ ut ⴱ+d dx ⳵ L ⳵ ux ⴱ− ⳵ L ⳵ uⴱ= ␦ F ␦ ut ⴱ,共16兲 with the dissipation function F=−i ␤ 共uut ⴱ−uⴱut兲,共17兲 and similarly for the complex conjugate of Eq. 共1兲. We note that more complicated dissipation terms than the simple term −i ␤ uin Eq. 共15兲can also be treated by generalizing the Euler-Lagrange formalism 关17兴. Now, inserting the ansatz Eq. 共4兲in Land F关32,33兴and integrating, we obtain L= 冕 −⬁ +⬁ dxL共u,uⴱ,ux,ux ⴱ,ut,ut ⴱ兲=L共Y ជ ;Y ជ ˙兲,共18兲 F= 冕 −⬁ +⬁ dxF共u,uⴱ,ux,ux ⴱ,ut,ut ⴱ兲=F共Y ជ ;Y ជ ˙兲.共19兲 Hence, the Mcollective coordinates satisfy Mgeneralized Lagrange equations d dt ⳵ L ⳵ Y ˙n − ⳵ L ⳵ Yn − ⳵ F ⳵ Y ˙n =0, 共20兲 or equivalently 冕 −⬁ +⬁ dx 冋 d dt ⳵ L ⳵ Y ˙n − ⳵ L ⳵ Yn − ⳵ F ⳵ Y ˙n 册 =0, 共21兲 where n=1,...,M. From now on we denote Y=Ynand assume 兵see Eq. 共4兲in 关32兴其 that 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 the following relations hold d dt ⳵ L ⳵ Y ˙=d dt 冉 ⳵ L ⳵ ut ⳵ u ⳵ Y 冊 +d dt 冉 ⳵ L ⳵ ut 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 ⳵ ut ⳵ ut ⳵ Y+ ⳵ L ⳵ ut ⴱ ⳵ ut ⴱ ⳵ Y,共25兲 ⳵ F ⳵ Y ˙= ⳵ F ⳵ ut ⳵ u ⳵ Y+ ⳵ F ⳵ ut ⴱ ⳵ uⴱ ⳵ Y,共26兲 and taking into account the relations Eqs. 共22兲and 共23兲, Eq. 共21兲becomes 冕 −⬁ +⬁ dx 再 d dt 冉 ⳵ L ⳵ ut 冊 +d dx 冉 ⳵ L ⳵ ux 冊 − ⳵ L ⳵ u− ⳵ F ⳵ ut 冎 ⳵ u ⳵ Y + 冕 −⬁ +⬁ dx 再 d dt 冉 ⳵ L ⳵ ut ⴱ 冊 +d dx 冉 ⳵ L ⳵ ux ⴱ 冊 − ⳵ L ⳵ uⴱ− ⳵ F ⳵ ut ⴱ 冎 ⳵ uⴱ ⳵ Y=0. 共27兲 Inserting the Lagrangian density Land the dissipation function Fin Eq. 共27兲, the first and second curly brackets become the NLS Eq. 共1兲and its corresponding complex conjugate equation, respectively 共up to a constant factor兲, where Ris given by Eq. 共15兲. Then, Eq. 共27兲represents the integration of the sum of the first NLSE Eq. 共1兲multiplied by ⳵ u/ ⳵ Yand the second NLSE, i.e., the complex conjugate of Eq. 共1兲, multiplied by ⳵ uⴱ/ ⳵ Y. This projection procedure is exactly the same as that which has produced the MEqs. 共5兲in Sec. II. Thus, the Lagrangian approach is equivalent to the GTWM and vice versa. However, in practice the GTWM requires more work if M⬎3, because M共M−1兲/2 integrals IYnYjand Mintegrals Rn共Y ជ 兲in Eqs. 共6兲and 共8兲, respectively, must be calculated. In the Lagrangian method one has to perform only M+3 integrals to obtain Land F. On the other hand, the GTWM is more general since the Lagrangian density of the perturbed system need not be known. IV. GTWM AND MODIFIED CONSERVED QUANTITIES A very peculiar property of GTWMs is related to its relationship with the so-called modified conservation laws 共MCL兲共the time evolution of the quantities which are conserved for the unperturbed system兲关18兴. Such an equivalence was already shown for NLKGEs in 关23兴since usually the GENERALIZED TRAVELING-WAVE METHOD,…PHYSICAL REVIEW E 82, 016606 共2010兲 016606-3 equations of motion for M CCs are related with the first M MCLs. In this section, we show that in the case of the NLSE the GTWM with the ansatz u共x,t兲=u共x− ␨ ,⌽, ␩ , ␰ 兲,uⴱ共x,t兲=uⴱ共x− ␨ ,⌽, ␩ , ␰ 兲, 共28兲 yields at first only 3 MCLs, namely the time variation of the norm, the momentum, and the energy. In Eq. 共28兲 ␨ denotes the soliton position and ⌽represents a phase, 共so that ⳵ u/ ⳵ ⌽=−iu and ⳵ uⴱ/ ⳵ ⌽=iuⴱ兲. We stress that we need not yet make a specific ansatz for u. With M=4 and setting n =1 in Eqs. 共5兲–共8兲we obtain 兺 j=1 4 I ␨ YjY ˙j=dP dt + 冕 −⬁ +⬁ dx ⳵ ⳵ x i 2关utuⴱ−uut 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 where Pis the momentum P= 冕 −⬁ +⬁ dx i 2关uux ⴱ−uⴱux兴.共32兲 Substituting Eqs. 共29兲–共31兲in Eq. 共5兲and taking into account that 冕 −⬁ +⬁ dx ⳵ ⳵ x 再 i 2关utuⴱ−uut ⴱ兴−H0 冎 =0, 共33兲 we obtain dP dt = 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ x+Rⴱ ⳵ u ⳵ x 冊 ,共34兲 which represents the variation of the momentum with respect to time for the perturbed NLSE Eq. 共1兲and its complex conjugate equation. Taking n=2 in Eqs. 共5兲–共8兲, we obtain 兺 j=1 4 I⌽YjY ˙j=−dN dt ,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, substituting Eqs. 共35兲–共37兲in Eq. 共5兲our second modified conservation law reads dN dt = 冕 −⬁ +⬁ dx i共Rⴱu−Ruⴱ兲,共38兲 and represents the variation of the norm N. Finally, we insert in Eqs. 共5兲–共8兲Y1= ␨ ,Y2=⌽,Y3= ␩ , and Y4= ␰ , with M=4. Then, we multiply these equation by ␨ ˙,⌽ ˙, ␩ ˙, and ␰ ˙, respectively; adding them yields 兺 n=1 4 兺 j=1 4 IYnYjY ˙jY ˙n=0, 共39兲 F1 ␨ ˙+F2⌽ ˙+F3 ␩ ˙+F4 ␰ ˙=−dH0 dt ,共40兲 R1 ␨ ˙+R2⌽ ˙+R3 ␩ ˙+R4 ␰ ˙=− 冕 −⬁ +⬁ dx共Rut ⴱ+Rⴱut兲,共41兲 so that the evolution of the energy is given by dH0 dt =− 冕 −⬁ +⬁ dx共Rut ⴱ+Rⴱut兲,共42兲 which can be written as dH0 dt =−兺 j=1 4 Y ˙j 冕 −⬁ +⬁ dx 冉 R ⳵ uⴱ ⳵ Yj +Rⴱ ⳵ u ⳵ Yj 冊 .共43兲 Equation 共43兲represents the third MCL for NLSE. We would like to find a fourth MCL, since we are using four CC equations from our GTWM. It is natural to expect that these equations are related not only to the first three conservation laws but also to the fourth one. The infinite sequence of conservation laws of the unperturbed NLSE can be obtained via the following recurrence relation for densities, which was obtained in the framework of the IST 关34兴, bm+1 =uⴱd dx 冉 bm uⴱ 冊 +兺 k+j=m−1 bkbj,共44兲 with b0=兩u兩2,m=0,1,....Asimilar relation, with bm+1 ⴱ, holds for the complex conjugate of Eq. 共1兲with R=0. In this way, for m=0 the momentum density P=−共i/2兲共b1−b1 ⴱ兲is obtained 关34兴, which is identical with that in Eq. 共32兲. For m =1, we obtain the energy density and for m=2 a new density, ␳ 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 the one-soliton solution for the unperturbed NLSE, i.e., Equation 共9兲with constant velocity and amplitude and QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲 016606-4 the linear functions in time ␨ = ␨ 0−4 ␰ tand ⌽=⌽0 −4共 ␰ 2+ ␩ 2兲t, this yields a new conserved quantity H1 =兰−⬁ +⬁dx ␳ 1共x,t兲=32 ␩ ␰ 共 ␰ 2− ␩ 2兲. Due to the perturbations R and Rⴱin Eq. 共1兲and its complex conjugate equation, respectively, the evolution of H1is determined by 关18兴 dH1 dt = 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 where H1is the Hamiltonian for the first integrable hierarchy of the NLSE 关35兴. Comparing the rhs of Eq. 共8兲with the rhs of Eq. 共46兲we realize that in the former equation the perturbation Ris multiplied by the first derivative of uⴱwith respect to a given CC, whereas in the latter one the third spatial derivative of uⴱappears. So, the MCL Eq. 共46兲is not equivalent to any equation or combination of equations of the system Eqs. 共5兲–共8兲obtained by means of GTWM. However, we can achieve our goal by considering the evolution of the first moment of the norm 关19兴, dN1 dt =2P+ 冕 −⬁ +⬁ dx x共Rⴱu−Ruⴱ兲,N1= 冕 −⬁ +⬁ dx x兩u兩2. 共47兲 We now show that Eq. 共47兲together with the first three MCLs yields the four CC Eqs. 共11兲–共14兲. Here we use the specific ansatz Eq. 共9兲for which the norm N=4 ␩ ,共48兲 the first moment of the norm, N1=4 ␩ ␨ ,共49兲 the momentum P=−8 ␩ ␰ ,共50兲 and the energy is given by Eq. 共10兲. Substituting Eqs. 共48兲–共50兲and 共10兲in the Eqs. 共38兲,共34兲,共47兲, and 共43兲, respectively, we obtain 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 From Eq. 共51兲and the identity ⳵ 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 provides the evolution equation for ␨ ˙. Taking into account 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 the evolution equation for ⌽since ␩ evolves in time. From Eqs. 共51兲and 共52兲we obtain the equation of motion for ␩ and ␰ . From Eqs. 共55兲and 共56兲, the equations for ␨ and ⌽can be obtained. In fact, these four equations agree with Eqs. 共11兲–共14兲obtained via GTWM. V. CONCLUSIONS We have developed the GTWM for the perturbed NLS Eqs. 共1兲and the complex conjugate of Eq. 共1兲共this is different from earlier applications of the method which started from the Hamilton equations for magnetic vortices and for solitons in NLKGEs兲. The GTWM is based on a projection technique: The NLSEs Eq. 共2兲and the complex conjugate of Eq. 共2兲are multiplied by derivatives of uⴱand uwith respect to one of the CCs, respectively. Then the resulting equations are added and integrated over the system. This yields the evolution equations for an arbitrary number of CCs, Eqs. 共5兲–共8兲, without using a specific ansatz. This method is more general than the variational approach because it is not necessary to know the Lagrangian density Lof the perturbed system. E.g., see Eq. 共1兲of 关36兴, where the Lagrangian density does not exist, but the Hamiltonian H0of the unperturbed system is known. In cases where Lexists and conditions Eqs. 共22兲and 共23兲are satisfied, we have shown that the GTWM and the variational formalism are equivalent, i.e., from both theories the same evolution equations for the CCs are obtained. Using 4 collective coordinates with the ansatz Eq. 共9兲, namely, the position ␨ , velocity ␰ , amplitude ␩ and phase ⌽ of the soliton, we have explicitly shown that the GTWM is equivalent to the results obtained from the IST, from the adiabatic perturbation theory and from the time variation of the norm, the first moment of the norm, the momentum, and the energy of the system. However, we stress that the GTWM is very concise, and more general than the IST and the time-variation of the conserved quantities since it can be applied when we have an ansatz with more than 4 CCs 共see the ansätze with 5 CCs in Refs. 关36,38兴for soliton propagation in optical fibers and with 6 CCs used in 关37兴兲. Moreover, the perturbations considered in Refs. 关36,38兴include several dissipative terms which present no additional difficulty for the GTWM, in contrast to the variational method. Work along this line is in progress. ACKNOWLEDGMENTS We thank Yuri Gaididei 共Kiev兲and Edward Arevalo 共Dresden兲for very useful discussions on this work. N.R.Q. GENERALIZED TRAVELING-WAVE METHOD,…PHYSICAL REVIEW E 82, 016606 共2010兲 016606-5 acknowledges financial support by the Ministerio de Educación y Ciencia 共MEC, Spain兲through Grant No. FIS2008- 02380/FIS, and by the Junta de Andalucía under Project Nos. FQM207, FQM-00481, P06-FQM-01735, and P09-FQM- 4643. F.G.M. acknowledges the hospitality of the University of Seville and of the Theoretical Division and Center for Nonlinear Studies at Los Alamos Laboratory. Work at Los Alamos was supported by U.S. DOE. F.G.M. acknowledges financial support by IMUS and by the Plan Propio of the University of Seville. 关1兴A. R. Bishop, J. A. Krumhansl, and S. E. Trullinger, Physica D 1,1共1980兲. 关2兴A. C. Scott, Nonlinear Science 共Oxford University, Oxford, 1999兲. 关3兴D. W. McLaughlin and A. C. Scott, Phys. Rev. A 18, 1652 共1978兲. 关4兴A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 共1998兲. 关5兴M. J. Rice and E. J. Mele, Solid State Commun. 35, 487 共1980兲. 关6兴M. Salerno and A. C. Scott, Phys. Rev. B 26, 2474 共1982兲. 关7兴V. Stehr, P. Müller, F. G. Mertens, and A. R. Bishop, Phys. Rev. E 79, 036601 共2009兲. 关8兴M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, Phys. Rev. B 15, 1578 共1977兲. 关9兴A. Bondeson, M. Lisak, and D. Anderson, Phys. Scr. 20, 479 共1979兲. 关10兴D. J. Kaup and A. C. Newell, Proc. R. Soc. London, Ser. A 361, 413 共1978兲. 关11兴D. J. Kaup, Phys. Rev. A 42, 5689 共1990兲. 关12兴J. Yan, Y. Tang, G. Zhou, and Z. Chen, Phys. Rev. E 58, 1064 共1998兲. 关13兴V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 46, 281 共1977兲. 关14兴Y. Kivshar and B. Malomed, Rev. Mod. Phys. 61, 763 共1989兲. 关15兴B. Malomed, in Progress in Optics, edited by E. Wolf 共North- Holland, Amsterdam, 2002兲, Vol. 43, p. 71. 关16兴A. Hasegawa, Chaos 10, 475 共2000兲. 关17兴S. Chávez Cerda, S. B. Cavalcanti, and J. M. Hickmann, Eur. Phys. J. D 1, 313 共1998兲. 关18兴V. I. Karpman, Phys. Scr. 20, 462 共1979兲. 关19兴W. L. Kath, Methods Appl. Anal. 4, 141 共1997兲. 关20兴F. G. Mertens, H. J. Schnitzer, and A. R. Bishop, Phys. Rev. B 56, 2510 共1997兲. 关21兴T. Kamppeter, F. G. Mertens, A. Sánchez, F. Domínguez- Adame, A. R. Bishop, and N. Gronbeck-Jensen, Eur. Phys. J. B7, 607 共1999兲. 关22兴T. Kamppeter, F. G. Mertens, E. Moro, A. Sánchez, and A. R. Bishop, Phys. Rev. B 59, 11349 共1999兲. 关23兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 62, 5695 共2000兲. 关24兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. Lett. 84, 871 共2000兲. 关25兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 62, R60 共2000兲. 关26兴L. Morales-Molina, N. R. Quintero, F. G. Mertens, and A. Sánchez, Phys. Rev. Lett. 91, 234102 共2003兲. 关27兴L. Morales-Molina, N. R. Quintero, A. Sánchez, and F. G. Mertens, Chaos 16, 013117 共2006兲. 关28兴A. V. Ustinov, C. Coqui, A. Kemp, Y. Zolotaryuk, and M. Salerno, Phys. Rev. Lett. 93, 087001 共2004兲. 关29兴G. P. Agrawal, Nonlinear Fiber Optics 共Academic, San Diego, 2001兲. 关30兴L. D. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Classics in Mathematics) 共Springer-Verlag, Berlin, 2007兲. 关31兴L. Morales-Molina, F. G. Mertens, and A. Sánchez, Phys. Rev. E72, 016612 共2005兲. 关32兴O. Legrand, Phys. Rev. A 36, 5068 共1987兲. 关33兴N. R. Quintero and E. Zamora-Sillero, Physica D 197,63 共2004兲. 关34兴S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method 共Springer-Verlag, Berlin, 1984兲. 关35兴A. Hasegawa and Y. Kodama, Solitons in Optical Communications 共Oxford, New York, 1995兲, pp. 70–72. 关36兴J. Santhanam and Govind P. Agrawal, Opt. Commun. 222, 413 共2003兲. 关37兴A. V. Maimistov, Sov. Phys. JETP 77, 727 共1993兲. 关38兴Z. Chen, A. J. Taylor, and A. Efimov, J. Opt. Soc. Am. B 27, 1022 共2010兲. QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲 016606-6