Full-time dynamics of modulational instability in spinor Bose-Einstein condensates
aa r X i v : . [ n li n . S I] A ug Full-time dynamics of modulational instability in spinor Bose–Einstein condensates
Evgeny V. Doktorov, ∗ Vassilis M. Rothos, and Yuri S. Kivshar B.I. Stepanov Institute of Physics, 68 Francisk Skaryna Avenue, 220072 Minsk, Belarus Department of Mathematics, Physics and Computational Sciences,Faculty of Technology, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Nonlinear Physics Centre and Australian Centre of Excellence for Quantum-Atom Optics,Research School of Physical Sciences and Engineering,Australian National University, Canberra ACT 0200, Australia
We describe the full-time dynamics of modulational instability in F = 1 spinor Bose–Einsteincondensates for the case of the integrable three-component model associated with the matrix non-linear Schr¨odinger equation. We obtain an exact homoclinic solution of this model by employingthe dressing method which we generalize to the case of the higher-rank projectors. This homoclinicsolution describes the development of modulational instability beyond the linear regime, and weshow that the modulational instability demonstrates the reversal property when the growth of themodulated amplitude is changed by its exponential decay. PACS numbers: 03.75.Lm, 03.75.Mn, 05.45.Yv
I. INTRODUCTION
Spinor Bose–Einstein condensate (BEC) with an opti-cal confinement represents a unique macroscopic systemwith the spin degrees of freedom [1, 2]. The interplaybetween the mean-field effective nonlinearities of three-component matter waves and their spin properties pro-duce many interesting phenomena such as the spin mix-ing [2], as well as the formation of spin domains [3, 4] andspin textures [5, 6]. Various properties of the spinor BEChave been analyzed theoretically [7, 8, 9]. The groundstate of the spinor BEC with the hyperfine spin F = 1can be either ferromagnetic (maximum spin projection)or polar (zero spin projection). It was shown in Ref. [10]within the linear stability analysis of the spinor conden-sate model that the ferromagnetic phase of the conden-sate can experience instability for large enough densitiesof atoms, while the polar phase remains always modula-tionally stable.Wadati and co-authors [11] demonstrated that thethree-component nonlinear equations describing the evo-lution of the F = 1 BEC can be reduced, under specialconstraints imposed on the condensate parameters, tothe completely integrable matrix nonlinear Schr¨odinger(NLS) equation [12]. Both bright and dark three-component BEC solitons have been found in the frame-work of this model [13, 14, 15, 16, 17].As regards the linear stability analysis presented inRef. [10], only an initial (linear) stage of the perturba-tion development can be explored by this method whichpredicts the exponential growth of the modulation fre-quency sidebands for some conditions, i.e., it describesthe conditions of modulational instability (MI). A physi-cal mechanism behind the MI is the parametric couplingbetween the spin degrees of freedom which leads to a pop- ∗ Electronic address: [email protected] ulation transfer between the spin components. To studythe long-time evolution of instabilities, numerical meth-ods are used as a rule. For the scalar NLS equation, theproblem of the long-time evolution of the MI was stud-ied by the truncation of the original model to a finitenumber of modes (as usually, the three-mode approxi-mation) [18]. More complete analysis of the long-timeMI dynamics [19, 20] is based on a linear constraint im-posed on the real and imaginary parts of solutions of thescalar NLS equation, and it allows one to find a class ofthree-parameter solutions sharing this property. Amongthe solutions found in such a way, a special solution de-scribes the development of MI beyond the linear regime,and it is identified as a homoclinic orbit separating twoqualitatively different types of periodic solutions. A sim-ilar result was obtained by means of the Darboux trans-formation with the plane wave as a ‘seed’ solution [21].Following terminology of Ref. [22], the full-time dynam-ics represents a nonlocal view of the MI development overa long time interval.A homoclinic orbit is a trajectory of a dynamical sys-tem that tends to the same manifold (fixed point, pe-riodic orbit, etc.) as time tends to ±∞ . The exis-tence of homoclinic solutions serves as an indicator ofchaotic regimes in a perturbed deterministic system. Fornonlinear wave systems described by partial differentialequations, the complete understanding of the homoclinicstructures in the infinite-dimensional phase space is farfrom being available at present. On the other hand, theunique features of the integrable nonlinear wave equa-tions admit essentially more deep insight into this prob-lem. Extended reviews of analytical and numerical meth-ods for obtaining homoclinic orbits for the scalar NLSand sine-Gordon equations are given in Refs. [23, 24].The aim of our paper is twofold. First, we derive ahomoclinic solution of the matrix NLS equation. Sec-ond, using these analytic results, we present the exactsolution of the problem of the long-time evolution of themodulationally unstable F = 1 BEC in the case when itis described by the integrable model.To find homoclinic solutions, we do not impose ad hoc constraints on the form of solutions. Instead, we use akind of dressing procedure, well known in the soliton the-ory [25], which was proposed recently as a systematic toolto generate exact homoclinic solutions of integrable non-linear equations with periodic boundaries [26]. A dress-ing factor being the main technical ingredient in this ap-proach contains a projector which determines the coor-dinate dependence of the homoclinic solution. It shouldbe pointed out that for all known homoclinic solutionsobtained up to now for various nonlinear equations (see,e.g., Refs. [26, 27]), this projector has rank 1. A cru-cial feature of the matrix NLS equation consists in thefact that the corresponding dressing factor incorporatesthe rank 2 projector. In terms of the soliton theory itcorresponds to multiple zeros of the scattering matrixcoefficients (or multiple zeros of the associated Riemann–Hilbert problem). Notice that the case of multiple zeroscannot be treated as a coalescence of simple zeros [28].Accordingly, we modify the definition of the dressing fac-tor for the case of the matrix NLS equation and obtainthe first example of the matrix homoclinic orbit and, asa result, the complete description of the MI evolution inthe integrable spinor BEC model.The paper is organized as follows. In Sec. II we de-scribe the integrable F = 1 BEC model. The methodfor obtaining homoclinic solutions for integrable nonlin-ear equations valid for higher rank projectors is outlinedin Sec. III. Section IV is devoted to the explicit deriva-tion of the homoclinic solution for the matrix NLS equa-tion and presents the main results of our paper. Thehomoclinic solution describes the temporal evolution of linearly unstable modes. We show that the MI has areversal property – the initial-wave profile is recoveredafter a sufficiently long time. Hence, the term ’side-bandinstability’ refers in fact to only the linear stage of theinstability development. Section V concludes the paper. II. MODEL
We consider an effective one-dimensional BEC trappedin a pencil-shaped region elongated in the x directionand tightly confined in the transversal directions. Theassembly of atoms in the hyperfine spin F = 1 stateis described by a vector order parameter ~ Φ( x, t ) =(Φ + ( x, t ) , Φ ( x, t ) , Φ − ( x, t )) T , where its components cor-respond to three values of the spin projection m F =1 , , −
1. The functions Φ ± and Φ obey a system ofcoupled Gross–Pitaevskii equations [13, 29] i ~ ∂ t Φ ± = − ~ m ∂ x Φ ± + ( c + c )( | Φ ± | + | Φ | )Φ ± + ( c − c ) | Φ ∓ | Φ ± + c Φ ∗∓ Φ , (2.1) i ~ ∂ t Φ = − ~ m ∂ x Φ + ( c + c )( | Φ + | + | Φ − | )Φ + c | Φ | Φ + 2 c Φ + Φ − Φ ∗ , where the constant parameters c = ( g + 2 g ) / c = ( g − g ) / g f ( f = 0 ,
2) is given in terms of the s -wave scat-tering length a f in the channel with the total hyperfinespin f , g f = 4 ~ a f ma ⊥ (cid:18) − C a f a ⊥ (cid:19) − . Here a ⊥ is the size of the transverse ground state, m isthe atom mass, and C = − ζ (1 / ≈ . c = c ≡ − c < . (2.2)The negative c means that we consider the ferromag-netic ground state of the spinor BEC with attractive in-teractions. The condition (2.2), being written in termsof g f as 2 g = − g >
0, imposes a constraint on thescattering lengths: a ⊥ = 3 Ca a / (2 a + a ). Redefin-ing the function ~ Φ as ~ Φ → ( φ + , √ φ , φ − ) T , normalizingthe coordinates as t → ( c/ ~ ) t and x → ( √ mc/ ~ ) x , andaccounting for the constraint (2.2), we obtain a reducedsystem of equations in a dimensionless form: i∂ t φ ± + ∂ x φ ± +2 (cid:0) | φ ± | + 2 | φ | (cid:1) φ ± +2 φ ∗∓ φ = 0 , (2.3) i∂ t φ + ∂ x φ +2 (cid:0) | φ + | + | φ | + | φ − | (cid:1) φ +2 φ + φ ∗ φ − = 0 . After arranging the components φ ± and φ into a 2 × Q = (cid:18) φ + φ φ φ − (cid:19) , we can easily see that Eqs.(2.3) take the form of the integrable matrix NLS equation i∂ t Q + ∂ x Q + 2 QQ † Q = 0 . (2.4)The matrix NLS equation (2.4) possesses the Lax repre-sentation with the 4 × U and V of the form[12] U = ik Λ + ˆ Q, Λ = diag( − , − , , , ˆ Q = (cid:18) Q − Q † (cid:19) , (2.5) V = 2 ik Λ + 2 k ˆ Q + i (cid:18) QQ † Q x Q † x − Q † Q (cid:19) , (2.6) k is a spectral parameter. III. METHOD
We are interested in periodic solutions of Eqs. (2.3)(or (2.4)) with a spatial period L , Q ( x + L, t ) = Q ( x, t ).Hence, the Floquet theory should be applied to analyzethe spectral problem M x = U M . (3.1)The fundamental solution M ( x, k ) of Eq. (3.1) is fixedby the condition M (0 , k ) = I , I is the unit 4 × T ( k ) as the fundamen-tal solution in the point x = L , T ( k ) = M ( L, k ). Di-agonalization of the transfer matrix determines a matrix R , R − T ( k ) R = diag (cid:0) e im L , . . . , e im L (cid:1) ≡ ∆( L, k ) , and produces the Floquet multipliers exp( im j L ) with theFloquet exponents m j , j = 1 , . . . ,
4. The Floquet spec-trum is a set of all k for which the transfer matrix T ( k )has the eigenvalues on the unit circle.The next step is a determination of a Bloch solution χ of Eq. (3.1) as χ = M R which obeys the property χ ( x + L, k ) = χ ( x, k )∆( L, k ), specific for the Bloch-typesolutions. The Bloch eigenfunctions of the periodic spec-tral problem (3.1) play the role of the Jost solutions ofthe spectral problem with a decreasing potential.Among the points of the Floquet spectrum we will dis-tinguish the so called double points [22]. Double pointsare those values of k for which the Floquet exponents m j differ in multipliers of 2 π/L or, in other words, theFloquet multipliers are degenerate. We will be especiallyinterested in complex double points which indicate lin-earized instability of solutions of Eq. (2.3) and label or-bits homoclinic to hyperbolic fixed points in the phasespace of a nonlinear system. Note that the term “dou-ble” in the context of the Floquet spectrum refers to thealgebraic multiplicity of a point of the spectrum and hasno relation to the multiplicity of zeros we have spokenabout in the Introduction. Real double points are asso-ciated with stable modes.Suppose we know explicitly a Bloch solution χ of thespectral problem χ x = U χ with the matrix U (2.5)whose entries contain the known solution Q of Eq. (2.4).Then we dress the solution χ by applying the dressingfactor D ( x, t, k ), χ = Dχ , and χ is a new solution ofthe spectral problem with new matrix U = DU D − + D x D − . The dressing factor has the form D = I − N X s =1 k s − k ∗ s k − k ∗ s P s ( x, t ) , (3.2)where P s is a projector, P s = P s , P s = 1 k s − k ∗ s r s X n,l =1 | n ; s i ( D ( s ) − ) nl h l ; s | ,D ( s ) nl = h n ; s | l ; s i k s − k ∗ s . (3.3)Here k s , s = 1 , . . . , N are complex double points of theFloquet spectrum and r s is the rank of the projector P s .The four-component ket- and bra-vectors | n ; s i and h l ; s | are the column and row arrays, respectively. Hence, | n ; s i is a four vector related with the s th complex double point k s and obtained by applying the Bloch solution χ ( k s )to a constant vector | q ; s i , | n ; s i = χ ( x, t, k s ) | q ; s i . (3.4) Exactly r ( s ) vectors | q ; s i , and hence r ( s ) vectors | n ; s i ,correspond to the complex double point k s . The sum-mation in Eq. (3.2) is taken over all N complex doublepoints, while that in Eq. (3.3) is performed over the r s -dimensional space of vectors | n ; s i produced in accor-dance with Eq. (3.4). Then a new solution of the matrixNLS equation is written asˆ Q = ˆ Q + N X s =1 ( k s − k ∗ s )[Λ , P s ] . (3.5)For the rank 1 projectors these formulas reduce to theknown ones [26].Note the essential difference in applications of thedressing procedure between the soliton theory and theperiodic wave theory. Indeed, the parameters k s are freein the standard use of the dressing method and, in anycase, they do note relate with the seed solution ˆ Q . Onthe contrary, our approach demands to choose k s as thecomplex double points of the Floquet spectrum of thespectral problem (3.1) for the seed solution χ .Hence, they are the complex double points k s and theprojectors P s that completely determine new solution.In the next section the above method will be used togenerate homoclinic solution of the spin 1 BEC model(2.3) and hence to reveal the long-time dynamics of theMI in this model. IV. RESULTS
We begin with a spatially homogeneous continuouswave solution of Eq. (2.3) with components φ (0)+ = φ (0) − = ae − iµt , φ (0)0 = ibe − iµt (4.1)as the seed solution to be dressed. Here a and b arereal constant amplitudes which determine a populationof each spin component, and the chemical potential µ isgiven by µ = − a + b ). Note the fixed π/ φ (0) ± and φ (0)0 . Thesame phase locking property is an inherent feature of thenonintegrable model (2.1) as well [10]. We could startwith more general representation of plane waves but thestructure of Eqs. (2.3) and the Galilean invariance re-duce it to the form (4.1). Then we consider the spectralproblem (3.1) with the matrix U containing the planewaves (4.1) as the potential Q : U = − ik ae − iµt ibe − iµt − ik ibe − iµt ae − iµt − ae iµt ibe iµt ik ibe iµt − ae iµt ik . (4.2)The fundamental solution of the spectral problem withthe matrix U is explicitly found: M = cos px + i ( k/p ) sin px a/p ) sin pxe − iµt i ( b/p ) sin pxe − iµt px + i ( k/p ) sin px i ( b/p ) sin pxe − iµt ( a/p ) sin pxe − iµt − ( a/p ) sin pxe iµt i ( b/p ) sin pxe iµt cos px − i ( k/p ) sin px i ( b/p ) sin pxe iµt − ( a/p ) sin pxe iµt px − i ( k/p ) sin px , det M = 1 , (4.3)where p = a + b + k . Diagonalization of the transfermatrix T ( k ) = M ( L, k ) is performed by the matrix R which has the form R = d − i ( a/b ) d [ b/ ( p + k )] d e − iµt − i [ a/ ( p + k )] d e − iµt − i ( a/b ) d d − i [ a/ ( p + k )] d e − iµt [ b/ ( p + k )] d e − iµt − [( p − k ) /b ] d e iµt d − [( p − k ) /b ] d e iµt d , (4.4)where d j are time dependent. As a result, R − T ( k ) R = ∆( L, k ) = diag( e − ipL , e − ipL , e ipL , e ipL ) . Therefore, the Floquet exponents are written as m = − p, m = − p, m = p, m = p (4.5)and have the multiplicity 2. Then we obtain the seedBloch solution χ = M R in the form χ = exp( i µ t ) d − i ab d bp + k d − iap + k d − i ab d d − iap + k d bp + k d k − pb d d k − pb d d × exp( ip Λ x + 2 ikp Λ t ) , (4.6)where the parameters d j ( t ) entering Eq. (4.4) have beendetermined from the second Lax equation χ t = V χ with the matrix V (2.6) depending on the seed continu-ous wave (4.1): d = d exp( − ik t ) , d = d exp(2 ik t ) ,d = d exp (cid:20) − i µt − ikpt (cid:21) , d = d exp (cid:20) i µt + 2 ikpt (cid:21) . Here d j are integration constants.Now we proceed to finding the complex double points.Following Ref. [22], we seek for double points as a differ-ence between two Floquet exponents m and m (4.5): m = m + δ s , δ s = 2 πL s, s = ± , ± , . . . . This gives k s = ± i p a + b − ( πs/L ) , if a + b > ( πs/L ) (4.7)and k s = ± p ( πs/L ) − ( a + b ) , if a + b < ( πs/L ) . (4.8)Hence, for given amplitudes a and b and period L thedouble points are arranged into infinite number of realdouble points (4.8) situated on the real axis in the k plane, and a finite number of complex double points(4.7) lying on the imaginary axis within the interval( − i √ a + b , i √ a + b ).Let us choose in the following the amplitudes and pe-riod in such a way that to obtain the single complexdouble point k (and hence − k ). It means √ a + b > ( π/L ) but √ a + b < (2 π/L ). For this choice the onlyrank 2 projector P ≡ P has the form P = 1 k − k ∗ X n,l =1 | n i ( D − ) nl h l | , D nl = h n | l i k − k ∗ . To simplify notations, we write | n i instead of | n ; 1 i . Since D is a 2 × P = 1 e D [ h | i| ih | − h | i| ih | − h | i| ih | + h | i| ih | ] , (4.9) e D = h | ih | i − h | ih | i , where the vectors | i and | i are determined as | i = χ ( k ) | q i , | i = χ ( k ) | r i . Here | q i and | r i are linearly independent constant vectorswith the components q j and r j , j = 1 , . . . ,
4. Then afterrather lengthy but straightforward algebraic calculationin accordance with Eqs. (4.9) and (3.5) taken for s = 1,we explicitly obtain new solutions of the integrable spin1 BEC model (2.3), φ + = φ − = ae − iµt (cid:18) i BA sin ψ (cid:19) , (4.10) φ = ibe − iµt (cid:18) i B A sin ψ (cid:19) , which at the same time represent components of the ma-trix homoclinic orbit of the matrix NLS equation (2.4).Here A = cosh 2 τ − cos 2 ρ sin ψ + 2 γ cosh τ sin ρ sin ψ + 12 γ , (4.11) B = sinh 2 τ cos ψ + i cosh 2 τ sin ψ − i cos 2 ρ sin ψ + 14 γ (cid:2) ( µ − e τ + µ ∗− e − τ ) e iρ − ( µ ∗ + e τ + µ + e − τ ) e − iρ (cid:3) + i γ (cid:18) sin ψ − ba cos ψ cos α (cid:19) , (4.12) B = sinh 2 τ cos ψ + i cosh 2 τ sin ψ − i cos 2 ρ sin ψ + 14 γ (cid:2) ( ν − e τ + ν ∗− e − τ ) e iρ − ( ν ∗ + e τ + ν + e − τ ) e − iρ (cid:3) + i γ (cid:16) sin ψ − ab cos ψ cos α (cid:17) . (4.13)The constants d j have been incorporated into the con-stant components q j and r j of the vectors | q i and | r i . Ifwe denote definite combinations of these components as e = q r − q r , e = q r − q r , e = q r − q r ,e = q r − q r , e j = | e j | e iα j , α jl = α j − α l , | e | = | e | , then the notations used in (4.11), (4.12) and (4.13) areas follows: k = ik , p = q a + b + k = πL , e iψ = p + ik √ a + b ,τ = 4 k p t + t , ρ = 2 p x − α , α = α ,e t = √ a + b b s | e || e | , γ = 2 | e | p | e || e | , - - - È Φ+ È - FIG. 1: Full-time evolution of the φ + (and φ − ) componentdue to modulational instability. The parameters are a = 1, b = 2, L = π/ α = π/ α = π/ | e j | = 1. µ ± = 1 ± ie ∓ iψ (cid:18) sin ψ − ba e iα cos ψ (cid:19) ,ν ± = 1 ± ie ∓ iψ (cid:16) sin ψ + ab e iα cos ψ (cid:17) . The solutions (4.10) are indeed homoclinic to the planewaves (4.1). Calculation of the asymptotics of φ ± and φ as t → ±∞ gives φ ± → ae − iµt e ± iψ , φ → ibe − iµt e ± iψ . In other words, these solutions reproduce in the limit t → ±∞ the seed plane waves up to a constant phase,as should be for the homoclinic orbit. Note that thenonlinear MI for the spin 1 condensate but for differentphases of the seed wave components was studied by theDarboux transformation in Ref. [14].Figures 1 and 2 illustrating the solution (4.10) demon-strate typical development of the continuous wave pertur-bation within three periods in x . We see that the stageof the exponential growth of instabilities revealed by thelinear stability analysis transforms to the exponential de-creasing with emergence of localized structures. Hence,the full-time evolution of MI for the integrable F = 1BEC model demonstrates the reversal property, such asthe Fermi–Pasta–Ulam process [30]: the phase trajectoryof the system returns to the initial one which correspondsto the continuous waves (4.1). For chosen parameters thegrowth and decrease development of the component φ is more pronounced than that of φ ± . V. CONCLUSIONS
We have derived the analytic formulas for describingthe full-time dynamics of the modulational instability in - - - È Φ o È - FIG. 2: Full-time evolution of the φ component due to modu-lational instability. The parameters are the same as in Fig. 1. the integrable model of F = 1 Bose–Einstein conden-sates. Our results are based on the exact homoclinicsolution of the matrix NLS equation with the continuousplane waves as an initial condition. We have shown thatthere exist cycles of the MI evolution with the reversalproperty when the exponential growth of the modula-tion amplitude changes to its exponential decay. Thesolution we present here is an example of large-amplitude periodic solutions. It describes an exponential growth of a weak modulation of a background for an initial stageof the condensate evolution, and in this sense the back-ground is unstable. However, in the nonlinear regime thisexponentially growing mode saturates and subsequentlytransforms into oscillations. As expected, the integrablemodel (2.3) does not exhibit long-time chaotic dynam-ics contrary to the regimes observed numerically for ageneral case [10], but it may serve as a good analyticalapproximation of the evolution of the condensate experi-encing the instability. Higher-order homoclinic solutionswhich correspond to several complex double points canbe obtained by the method described in [26].Strictly speaking, the analysis based on the continu-ous wave model is not applicable to the trapped systems.Nevertheless, such an approach remains valid when thetypical spatial extent of the condensate is larger thanthe period of the localized pattern formed in result ofthe instability. More realistic models should account fora (small) deviation of the condensate parameters fromthe constraint which provides integrability of the model.There exists an approach [31] to reveal a persistenceof the homoclinic orbit when the integrability conditionbreaks, and therefore to establish analytically the exis-tence of chaos. This approach is based on the construc-tion of the so-called Melnikov function from the quadraticproducts of the Bloch functions evaluated on the homo-clinic orbit. In this paper we have explicitly built allthe ingredients to perform the Melnikov analysis. Corre-sponding results will be published elsewhere. [1] D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S.Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys.Rev. Lett. , 2027 (1998).[2] M.-S. Chang, C.D. Hamley, M.D. Barrett, J.A. Sauer,K.M. Fortier, W. Zhang, L. You, and M.S. Chapman,Phys. Rev. Lett. , 140403 (2004).[3] H.J. Miesner, D.M. Stamper-Kurn, J. Stenger, S. Inouye,A.P. Chikkatur, and W. Ketterle, Phys. Rev. Lett. ,2228 (1999).[4] T. Isoshima, K. Machida, and T. Ohmi, Phys. Rev. A , 4857 (1999).[5] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. , 1822(1998).[6] A.E. Leanhardt, Y. Shin, D. Kielpinski, D.E. Pritchard,and W. Ketterle, Phys. Rev. Lett. , 140403 (2004).[7] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[8] H. Pu, C.K. Law, S. Raghavan, J.H. Eberly, and N.P.Bigelow, Phys. Rev. A , 1463 (1999).[9] M. Ueda and M. Koashi, Phys. Rev. A , 063602 (2002).[10] N.P. Robins, Weiping Zhang, E.A. Ostrovskaya, and Yu.S. Kivshar, Phys. Rev. A , 021601(R) (2001).[11] J. Ieda, T. Miyakawa, and M. Wadati, Phys. Rev. Lett. , 194102 (2004).[12] T. Tsuchida and M. Wadati, J. Phys. Soc. Jpn. , 1175(1998).[13] J. Ieda, T. Miyakawa, and M. Wadati, J. Phys. Soc. Jpn. , 2996 (2004). [14] L. Li, Z. Li, B.A. Malomed, D. Mihalache, and W.M.Liu, Phys. Rev. A , 033611 (2005).[15] M. Wadati and N. Tsuchida, J. Phys. Soc. Jpn. ,014301 (2006).[16] M. Uchiyama, J. Ieda, and M. Wadati, J. Phys. Soc. Jpn. , 064002 (2006)[17] J. Ieda, M. Uchiyama, and M. Wadati, J. Math. Phys. , 013507 (2007).[18] S. Trillo and S. Wabnitz, Opt. Lett. , 986 (1991).[19] N.N. Akhmediev, V.M. Eleonsky, and N.E. Kulagin, Zh.Exp. Teor. Fiz. , 1542 (1985).[20] N.N. Akhmediev and A. Ankiewicz, Solitons: NonlinearPulses and Beams (Chapman and Hall, London, 1997).[21] A.R. Its, A.V. Rybin, and M.A. Salle, Teor. Mat. Fiz. , 29 (1998).[22] M.G. Forest, D.W. MacLaughlin, D.J. Muraki, and O.C.Wright, J. Nonlinear Sci. , 291 (2000).[23] D.M. McLaughlin and E.A. Overman II, Surveys in Appl.Math. , 83 (1995).[24] M.J. Ablowitz and B.M. Herbst, SIAM J. Appl. Math. , 339 (1990); M.J. Ablowitz, B.M. Herbst, and C.M.Schober, J. Phys. A , 10671 (2001).[25] S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E.Zakharov, Theory of Solitons, the Inverse ScatteringMethod (Consultant Bureau, New York, 1984).[26] E.V. Doktorov and V.M. Rothos, Phys. Lett. A , 59(2003); E.V. Doktorov and V.M. Rothos, in:
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