Xing Lü a,, Hong-Wu Zhu a, Xiang-Hua Meng a, Zai-Chun Yang a, Bo Tian a,b

Transcription

1 J. Math. Anal. Appl. 336 (2007) Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications Xing Lü a,, Hong-Wu Zhu a, Xiang-Hua Meng a, Zai-Chun Yang a, Bo Tian a,b a School of Science, PO Box 22, Beijing University of Posts and Telecommunications, Beijing 00876, China b Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 00876, China Received 3 January 2007 Available online 3 March 2007 Submitted by C. Rogers Abstract Under investigation in this paper is a generalized nonlinear Schrödinger model with variable dispersion, nonlinearity and gain/loss, which could describe the propagation of optical pulse in inhomogeneous fiber systems. By employing the Hirota method, one- and two-soliton solutions are obtained with the aid of symbolic computation. Furthermore, a general formula which denotes multi-soliton solutions is given. Some main properties of the solutions are discussed simultaneously. As one important property of nonlinear evolution equation, the Bäcklund transformation in bilinear form is also constructed, which is helpful on future research and as far as we know is firstly proposed in this paper Elsevier Inc. All rights reserved. Keywords: Variable-coefficient nonlinear Schrödinger equation; Bilinear form; Multi-soliton solutions; Bäcklund transformation; Symbolic computation This work has been supported by the Key Project of Chinese Ministry of Education (o ), by the Specialized Research Fund for the Doctoral Program of Higher Education (o ), Chinese Ministry of Education, and by the ational atural Science Foundation of China under Grant o * Corresponding author. address: xinglv655@yahoo.com.cn (X. Lü) X/$ see front matter 2007 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 306 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) Introduction During the past decades, investigations on the optical fiber communications have become more and more attractive,2], in which the optical solitons have their potential applications in optical fiber transmission systems 2 5]. As we all know, solitonic structures are seen in many fields of sciences and engineering 6 9], among which an optical soliton exists in a fiber on the basis of the exact balance between the group velocity dispersion (GVD) and the self-phase modulation (SPM). The propagation of the optical solitons is usually governed by the nonlinear Schrödinger equation (LSE), which is one of the most important models in modern nonlinear science. Moreover, much attention has been paid to the investigation on the generalized LSEs with constant coefficients as a kind of ideal models of the much more complicated physical problems 0,]. As a matter of fact, in a real fiber exist some fiber nonuniformities to influence various effects such as the gain or loss, GVD and SPM, etc. Considering the varying dispersion, nonlinearity and gain/loss, we would like to investigate the following generalized variable-coefficient LSE 4]: iu x b(x)u tt l(x) u 2 u id(x)u = 0, () where u(x, t) is the complex envelope of the electrical field in the comoving frame, x is the propagation distance, t is the retarded time and the subscripts denote partial derivatives. All the parameters b(x), l(x) and d(x) are real analytic functions of the propagation distance x, which represent the GVD, SPM and distributed gain/loss, respectively. The equation describes the amplification or attenuation of pulse propagation in a single-mode optical fiber with distributed dispersion and nonlinearity. For describing different models in the optical fiber systems, two main cases of Eq. () are written as follows:. iu x B (x)u tt L u 2 u id (x)u = 0, (2) with L = or 2. For L =, the chromatic dispersion B (x) normalized to the standard monomode fiber dispersion coefficient and D (x) related to the nonlinear length, fiber loss and the amplifier location, Eq. (2) describes the averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, which has been derived and researched in Ref. 2]. For L = 2 and D (x) reduces to the normalized amplitude loss, which is related to the fiber power loss α, through the relation D (x) = αl D (L D is the characteristic dispersive length), the effect of different D (x) on the mode conversion in the intermediate fiber using a soliton input pulse has been discussed detailedly in Ref. 3]. In Ref. 4], the exact soliton solutions of the averaged dispersion-managed soliton system equation have been derived 5]. 2. iu x B 2 (x)u tt L (x) u 2 u = 0, (3) which describes the pulse propagation in the weakly nonlinear dispersive media 3] with L (x) = 2. The parameter B 2 (x) represents the normalized dispersion; in the case of propagation in a single fiber B 2 (x) =, while in the case of propagation through a fiber junction composed of three different fibers B 2 (x) is the step-like dispersion in the fibers relative to the last fiber. By the way, when B 2 (x) is the stepwise function describing the dispersion map, the propagation of dispersion-managed dark soliton is governed by Eq. (3) for a dimensionless envelope of the electrical field 6]. When the functions B 2 (x) and L (x) are periodic, B 2 (x) = B 2 (x ) and

3 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) L (x) = L (x ) and the x scale is made dimensionless by using a common period x 0 of the dimensional functions B 2 (x) and L (x), the parameter B 2 (x) may describe a fast variation of the local dispersion. The variation of the parameter L (x) is, for example, motivated by driving and damping in the original system 7,8]. Much work has been done about Eq. () for its important practical applications. A procedure based on the Bogolyubov method has been proposed and as a result, the asymptotic equation in the dominating order was obtained 8], the multiple-scale method was applied to study the dispersion-managed soliton power enhancement 9], the guiding-center theory based on the Lie transform was developed 20] but without a detailed analysis of different limits for the coefficients, the self-similarity technique was used for finding exact self-similar solutions describing both periodic and solitary waves ], the Darboux transformation based on the Lax Pair was presented through which soliton solutions were generated 2] and under certain constraints, four transformations from Eq. () to standard and cylindrical LSEs have been constructed 7]. Here we mainly use the Hirota method to study Eq. (). The outline of this paper is as follows. In Section 2, we will transform Eq. () into the bilinear form and obtain soliton solutions with symbolic computation 6 8]. The properties of the obtained solutions are also analyzed. In Section 3, we will derive the bilinear Bäcklund transformation for Eq. (). Section 4 will be our conclusions. 2. Bilinear form and soliton solutions It is well known that the Hirota method 22] is a straightforward and useful tool for dealing with soliton problems for nonlinear evolution equations. One definite advantage of this method is that if we once transform the original equation into the bilinear form, we can find solutions directly by the perturbation technique without employing the inverse scattering method. Equation () can be transformed into the variable-coefficient bilinear form idx b(x)dt 2 ] (g f)= 0, (4) D 2 t (f f)= a g 2, l(x) with a = C 2 b(x) e 2 d(x)dx, by the following transformation: u(x, t) = C e d(x)dx g(x,t) f(x,t), (6) where C 0 is an arbitrary real constant of integration, g(x,t) and f(x,t) are taken to be complex and real differentiable functions, respectively, and D x and Dt 2 are the bilinear derivative operators 23] defined by ( Dx m Dn t (f g) = x ) m ( x t ) n t f(x,t)g(x,t ). x =x,t =t Generally speaking, Eq. () is not integrable. To solve Eq. (), we consider the following constraint: l(x) = C 0 b(x)e 2 d(x)dx, which is a condition for Eq. () to pass the Painlevé test 24,25]. With the constraint (7), the parameter a in Eq. (5) reduces to C 0 C 2. For obtaining the soliton solutions of Eq. () by solving (5) (7)

4 308 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) Eqs. (4) and (5), without loss of generality, here we set a = C 0 C 2 =. Expand f(x,t) and g(x,t) as the following forms: f(x,t)= ε 2 f 2 (x, t) ε 4 f 4 (x, t) ε 6 f 6 (x, t), (8) g(x,t) = εg (x, t) ε 3 g 3 (x, t) ε 5 g 5 (x, t), (9) where the coefficients f n (x, t) and g m (x, t) for n = 2, 4, 6,...and m =, 3, 5,..., are the functions to be determined. Substituting expressions (8) and (9) into Eqs. (4) and (5) and collecting the coefficients of the same power of ε, wehave ε 0 : Dt 2 ( ) = 0, ε ( : idx b(x)dt 2 ) (g ) = 0, ε 2 : Dt 2 ( f 2 f 2 ) = g g, ε 3 ( : idx b(x)dt 2 ) (g f 2 g 3 ) = 0, ε 4 : Dt 2 ( f 4 f 2 f 2 f 4 ) = g g3 g 3g,. where denotes complex conjugate. In order to obtain the one-soliton solution of Eq. (), let us choose g (x, t) = e θ with θ = ηt ψ(x) η 0, where η and η 0 are complex constants and ψ(x) is a differentiable function to be determined. Via symbolic computation, ψ(x) and f 2 (x, t) are determined to be ψ(x)= iη 2 b(x)dx, f 2 (x, t) = 2(η η ) 2 eθθ. Hereby we can truncate the expansion with f i (x, t) = 0 (i = 4, 6, 8,...) and g j (x, t) = 0 (j = 3, 5, 7,...). Consequently, without loss of generality, setting ε =, f(x,t) and g(x,t) can be reduced to f(x,t)= 2(η η ) 2 eθθ, g(x,t)= e θ with θ = ηt iη 2 b(x)dx η 0. Then the one-soliton solution of Eq. () in explicit form is generated u(x, t) = C e d(x)dx e 2(ηη ) θθ 2 2b(x) l(x) Re(η) eiφ sech θ 0, C > 0, = 2b(x) l(x) Re(η) eiφ sech θ 0, C < 0, where φ = Im(θ), θ 0 = 2 θ θ ln( )]. 2(ηη ) 2 For constructing the two-soliton solution, we choose e θ g (x, t) = e θ e θ 2 with θ = η t iη 2 b(x)dx η 0 and θ 2 = η 2 t iη2 2 b(x)dx η2 0, (0)

5 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) where η,η 2,η 0,η0 2 are all complex constants. With the help of symbolic computation, f(x,t) and g(x,t) are obtained as f(x,t) = 2(η η )2 eθ θ 2(η 2 η2 )2 eθ 2θ2 2(η η2 )2 eθ θ2 2(η 2 η )2 eθ 2θ (η η 2 ) 2 (η η 2 )2 4(η η )2 (η η2 )2 (η 2 η )2 (η 2 η2 )2 eθ θ 2 θ θ 2, g(x,t) = e θ e θ 2 (η η 2 ) 2 2(η η )2 (η 2 η )2 eθ θ 2 θ (η η 2 ) 2 2(η η2 )2 (η 2 η2 )2 eθ θ 2 θ2, and the two-soliton solution is yielded u(x, t) = C e d(x)dx e θ e θ 2 (η η 2 ) 2 2(η η )2 (η 2 η )2 eθ θ 2 θ (η η 2 ) 2 2(η η 2 )2 (η 2 η 2 )2 eθ θ 2 θ 2 ] / 2(η η )2 eθ θ 2(η 2 η2 )2 eθ 2θ2 2(η η2 )2 eθ θ2 2(η 2 η )2 eθ 2θ (η η 2 ) 2 (η ] η 2 )2 4(η η )2 (η η2 )2 (η 2 η )2 (η 2 η2 )2 eθ θ 2 θ θ 2 b(x) =± e (θ θ 2 δ 2 ) 2 cosh ɛ 2 e (θ 2 θ δ ) ]/ cosh(ɛ ɛ 2 σ ) 2 cosh ɛ l(x) 2(η η )(η 2 η2 ) cosh ɛ 3 2(η η2 )(η 2 η ) (η η 2 )(η η 2 ) cosh(ɛ ɛ 2 σ 2 ) 2(η η )(η η2 )(η 2 η )(η 2 η2 ) () (if C > 0, the sign on the right-hand side of Eq. () is taken to be positive; otherwise negative) with η2 η ] η σ = ln η η2, σ 2 = ln η2 ], η η 2 (η η 2 ) 2 ] (η η 2 ) 2 ] δ = ln 2(η η )(η 2 η, δ 2 = ln )2 2(η η2 )(η 2 η2, )2 ɛ = θ θ 2( δ ), ɛ2 = ( θ2 θ2 2 δ ) 2, ɛ3 = ( θ θ 2 θ 2 θ ) 2 σ. With the variable coefficients l(x), b(x) and d(x) satisfying constraint (7), the -soliton solution of Eq. () in the sense of Ref. 26] can be expressed in the following form: u(x, t) = C e d(x)dx g(x,t) f(x,t), where ]

6 30 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) f(x,t)= exp v=0, g(x,t) = exp v=0, g (x, t) = exp v=0, ( 2 i<j ( 2 i<j ( 2 i<j ξ(i,j)v i v j ξ(i,j)v i v j ξ(i,j)v i v j v i θ i ), v i θ i ), v i θ i ), where θ i = η i t iηi 2 b(x)dx ηi 0, for i =, 2,...,2; θ i = θi, η i = ηi, for i =, 2,...,; C 2 ξ(i,j) = ln l(x) ] 2(η i η j ) 2 b(x)e 2, d(x)dx for i =, 2,..., and j =,...,2, or i =,...,2 and j =, 2,...,; C 2 ξ(i,j) = ln l(x) ] 2(η i η j ) 2 b(x)e 2, d(x)dx for i =, 2,..., and j =, 2,...,, or i =,...,2 and j =,...,2, where η i and ηi 0 are different complex constants related, respectively, to the amplitude and the phase of the ith soliton, 2 i<j indicates the summation over all possible pairs taken from 2 elements with the specified condition i<j, and v=0,, v=0, and v=0, indicate the summations over all possible combinations of v i = 0, (i =, 2,...,2), respectively, under the conditions v i = v i, v i = v i and v i = From expression (0), it is worth noting that the pulse width (amplitude) and the frequency shift (group-velocity) are determined by the real and imaginary part of η, and the initial position and the initial phase are related to the real and imaginary part of η 0, respectively. Here we provide some figures to describe the soliton solutions obtained by the Hirota method. Some solitons here, for example plotted in Fig. 3, are similar to these in Ref. 27] and the distinction lies in that the variable coefficients are taken differently there but they are coincident with each other. Figure shows the shape and motion of the one-soliton solution given by expression (0) when the variable coefficients are taken to be some different constants. For describing the varying group velocity dispersion, nonlinearity and gain/loss, we would like to consider some variable-coefficient cases. Figure 2 describes the amplitude of the one-soliton solution moving with periodic growth and decay with the given parameters. Figure 3 depicts the propagation of the one-soliton solution with periodic oscillation along the distance x. In Fig. 4 we take a periodic distributed amplification system with the varying group velocity dispersion parameter and nonlinearity parameter as an example. From it we can clearly see that the soliton keeps its shape even if the group velocity is changed, which is one of the important properties of solitons. Figure 5 presents the separating evolution plot of the two-soliton solution given by expression (). v i.

7 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) (a) (b) Fig.. The parameters adopted here are: b(x) =, d(x) = 0, l(x) =, C =, C 0 =. (a) η = 0.2i, η 0 = 2i; (b) η =.4 0.i, η 0 = 3 3i. (a) (b) Fig. 2. The parameters adopted here are: η =.8, η 0 = 0.05, b(x) =, C =, C 0 =. (a) d(x) = 0.2sin(x), l(x) = exp( 0.4cos(x));(b)d(x) = 0.2cos(x), l(x) = exp(0.4sin(x)). (a) (b) Fig. 3. The parameters adopted here are: d(x) = 0, C =, C 0 =, η =.8 0.6i, η 0 = 0. (a) b(x) = 0.5sin(0.5x), l(x) = 0.5sin(0.5x);(b)b(x) = 0.5cos(0.5x), l(x) = 0.5cos(0.5x). From this figure we notice that the separation between two solitons keeps constant. Figure 6 shows the interaction of two-soliton solutions. It can be noted that two solitons collide elastically without perturbation, that is to say, after collision the two-soliton waves maintain their original shapes and velocities with only a phase shift at the moment of collision. This is also one of the important properties of solitons.

8 32 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) (a) (b) Fig. 4. The parameters adopted here are: l(x) = 0.5cos(x), C =, C 0 =, η =.8 0.6i, η 0 = 0. (a) b(x) = 0.5exp( 0.02x)cos(x), d(x) = 0.0; (b) b(x) = 0.5exp(0.02x)cos(x), d(x) = 0.0. Fig. 5. The parameters adopted here are: b(x) = 0.5exp(0.05x)cos(x), l(x) = 0.5cos(x), d(x) = 0.025, C =, C 0 =, η =.2 0.7i, η 2 =. 0.7i, η 0 = i, η0 2 = i. Fig. 6. The parameters adopted here are: b(x) = 0.5cos(0.5x), l(x) = 0.5cos(0.5x), d(x) = 0, C =, C 0 =, η =.5 0.8i, η 2 = 2.2 2i, η 0 = i, η0 2 = 2 2i. 3. Bilinear Bäcklund transformation The Bäcklund transformation (BT) plays an important role in the studies of various solitons 8,28]. It provides a means of constructing new solutions from known solutions to soliton equations. In this section we will derive a bilinear BT for Eq. () from Eqs. (4) and (5). In order to obtain a BT between (f, g) and (f,g ), we consider P ( id x b(x)dt 2 ) g f ] f 2 f 2( id x b(x)dt 2 ) ] g f = 0. (2) By means of some identities about the bilinear derivative operators and using Eq. (5), Eq. (2) may be rewritten as P id x (g f f g)f f (g f f g) id x (f f) ]

9 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) By taking 2b(x)D t Dt (g f f g) f f ] b(x)(g f f g)dt 2 (f f) b(x)dt 2 (g f f g)f f 2 C2 e 2 d(x)dx l(x) (g f f g) ( g g g g ) (g f f g) ( g g g g )] = 0. (3) D t (g f f g) = μ(x)(g f f g), (4) Eq. (3) can be decoupled into b(x)d 2 t 2μ 2 (x)b(x) ] f f C 2 e 2 d(x)dx l(x)re g g ] = 0, (5) idx λ(x, t) ] (g f f g) b(x)d 2 t (g f f g) = 0, (6) idx 2μ(x)b(x)D t λ(x, t) ] f f ic 2 e 2 d(x)dx l(x)im g g ] = 0, (7) where μ(x) is a function of the variable x and λ(x, t) is a function of the variables x and t. Thus, the set of Eqs. (4) (7) constitutes a BT for Eq. (). Introducing the vacuum solution g(x,t) = 0, f(x,t)=, namely, u(x, t) = 0, into Eqs. (4) (7), we obtain g (x, t) t = μ(x)g (x, t), f (x, t) tt 2μ 2 (x)f (x, t) = 0, ig (x, t) x λ(x, t)g (x, t) b(x)g (x, t) tt = 0, if (x, t) x 2μ(x)b(x)f (x, t) t λ(x, t)f (x, t) = 0, where the subscripts t and x denote partial derivatives. Solving these linear differential equations above, under the constraint (7), we get the one-soliton solution of Eq. () e iktik2 b(x)dxk0 iφ 0 u (x, t) = C e d(x)dx e 2kt 2 2k 2 b(x)dxδ 0 e 2kt2 2k 2 b(x)dx δ 0 C 0 b(x) = C e iφ sech θ l(x) 0, (8) provided μ(x) = ik, λ(x, t) = 0, where k 0, φ 0 and k 0 are arbitrary real constants, δ 0 is an arbitrary constant of integration, C = C 2 e k 0, φ = kt k2 b(x)dx φ 0, and θ 0 = 2kt 2 2k 2 b(x)dx δ 0. With the help of symbolic computation, it can be proved that, under the constraint (7), u (x, t) corresponding to expression (0) is indeed a solution of Eq. (). 4. Conclusions The optical solitons are of great interest theoretically and experimentally for their potential applications in the optical fiber transmission system. With the consideration of varying dispersion, nonlinearity and gain/loss, Eq. () describes the propagation of optical pulse in inhomogeneous fiber. In practical applications, the model is of primary interest not only for the compression and amplification of optical solitons in inhomogeneous systems, but also for the stable transmission of soliton control. From the integrable point of view, that is, under the constraint (7), some soliton solutions are derived and investigated in Section 2. Some figures are plotted to illustrate the properties of the solutions. The properties are meaningful for the investigation on the stability of soliton propagation in the optical soliton communications.

10 34 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) Moreover, in this paper, we have derived a bilinear BT for Eq. (). As an important property of nonlinear evolution equation, it has not been given before. Via the bilinear BT, we have obtained a solution of Eq. (), i.e., expression (8). According to expression (8), it is worth noting that the pulse width and the group-velocity (phase shift) are related to the parameter k, and the initial position and the initial phase of soliton are determined by the parameters δ 0 and φ 0, respectively. The soliton amplitude is given by u (x, t) = C C0 b(x) l(x) sech θ 0 and the maximum amplitude is C C0 b(x) l(x), which depends on C 0, C, k 0, k, b(x) and l(x). Special attention should be paid to that the constraint for expression (8) to be a solution of Eq. () is the same as constraint (7) which is given by the Painlevé test. Thus, we can predict that constraint (7) is a condition under which Eq. () is completely integrable. The work carried out in this paper may be useful for further study. Acknowledgments We express our sincere thanks to Professor Y.T. Gao for his valuable comments. We also would like to thank Ms. L.L. Li, Ms. J. Li and Mr. H.Q. Zhang for their helpful suggestions. References ] G.P. Agrawal, onlinear Fiber Optics, Academic Press, San Diego, 995; G.I. Stegeman, D.. Christodoulies, M. Segev, IEEE J. Select. Top. Quant. Electron. 6 (2000) 49; B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B Quantum Semiclass. Opt. 7 (2005) R53. 2] B. Tian, Y.T. Gao, Phys. Lett. A 342 (2005) 228; B. Tian, Y.T. Gao, Phys. Lett. A 359 (2006) 24; B. Tian, Y.T. Gao, H.W. Zhu, Phys. Lett. A (2007), in press, doi:0.06/j.physleta ] B.A. Malomed, Opt. Lett. 9 (994) 34; M. akazawa, H. Kubota, K. Suzuki, E. Yamada, Chaos 0 (2002) 486; V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. Lett. 90 (2003) 3902; L.Y. Wang, L. Li, Z.H. Li, G.S. Zhou, Phys. Rev. E 72 (2005) ] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, W.J. Tomlinson, Opt. Lett. 8 (983) ] H.W. Zhu, B. Tian, Sciencepaper Online, Chinese Ministry of Education, o , cn/index.html. 6] G. Das, J. Sarma, Phys. Plasmas 6 (999) 4394; M.P. Barnett, J.F. Capitani, J. Von Zur Gathen, J. Gerhard, Int. J. Quantum Chem. 00 (2004) 80; F.D. Xie, X.S. Gao, Commun. Theor. Phys. 4 (2004) 353; B. Li, Y. Chen, H.. Xuan, H.Q. Zhang, Appl. Math. Comput. 52 (2004) 58; Y.T. Gao, B. Tian, Phys. Plasmas 3 (2006) 290; Y.T. Gao, B. Tian, Phys. Plasmas (Lett.) 3 (2006) 20703; Y.T. Gao, B. Tian, Phys. Lett. A 349 (2006) 34; B. Tian, G.M. Wei, C.Y. Zhang, W.R. Shan, Y.T. Gao, Phys. Lett. A 356 (2006) 8. 7] Z.Y. Yan, H.Q. Zhang, J. Phys. A 34 (200) 785; W.P. Hong, Phys. Lett. A 36 (2007) 520; B. Tian, Y.T. Gao, Phys. Lett. A 340 (2005) 243; B. Tian, Y.T. Gao, Phys. Lett. A 340 (2005) 449; B. Tian, Y.T. Gao, Phys. Plasmas 2 (2005) 05470; B. Tian, Y.T. Gao, Phys. Lett. A 362 (2007) ] B. Tian, Y.T. Gao, Phys. Plasmas (Lett.) 2 (2005) ; B. Tian, Y.T. Gao, Eur. Phys. J. D 33 (2005) 59; Y.T. Gao, B. Tian, Europhys. Lett. 77 (2007) 500; Y.T. Gao, B. Tian, Phys. Lett. A 36 (2007) ] Z.Z. Yao, C.Y. Zhang, H.W. Zhu, X.H. Meng, T. Xu, B. Tian, Sciencepaper Online, Chinese Ministry of Education, o ,

11 X. Lü et al. / J. Math. Anal. Appl. 336 (2007) ] Y.S. Kivshar, V.V. Afanasjev, Phys. Rev. A 44 (99) R446; M. Gedalin, T.C. Scott, Y.B. Band, Phys. Rev. Lett. 78 (997) 448; Q.H. Park, H.J. Shin, Phys. Rev. Lett. 82 (999) 4432;.C. Panoiu, D. Mihalache, D. Mazilu, L.C. Crasovan, I.V. Mel nikov, F. Lederer, Chaos 0 (2000) 625; V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. E 7 (2005) ] D. Mihalache, L. Torner, F. Moldoveanu,.C. Panoiu,. Truta, Phys. Rev. E 48 (993) ] I.R. Gabitov, S.K. Turitsyn, Opt. Lett. 2 (996) ] K. Bertilsson, T. Aakjer, M.L. Quiroga-Teixeiro, P.A. Andrekson, P.O. Hedekvist, Opt. Fiber Technol. (995) 7. 4] Z.Y. Xu, L. Li, Z.H. Li, G.S. Zhou, K. akkeeran, Phys. Rev. E 68 (2003) ] A. Hasegawa, Y. Kodama, A. Maruta, Opt. Fiber Technol. 3 (997) 97. 6] H. Li, D.X. Huang, Chin. Phys. Lett. 3 (2003) 47. 7] B. Tian, W.R. Shan, C.Y. Zhang, G.M. Wei, Y.T. Gao, Eur. Phys. J. B (Rapid ot.) 47 (2005) ] A. Wingen, K.H. Spatschek, S.B. Medvedev, Phys. Rev. E 68 (2003) ] T.S. Yang, W.L. Kath, Opt. Lett. 22 (997) 985; T.S. Yang, W.L. Kath, S.K. Turitsyn, Opt. Lett. 23 (998) 597; T.S. Yang, W.L. Kath, S.K. Turitsyn, J. Engrg. Math. 36 (999) ] A. Hasegawa, Y. Kodama, Opt. Lett. 6 (99) 385; A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Clarendon Press, Oxford, 995; I. Gabitov, T. Schäfer, S.K. Turitsyn, Phys. Lett. A 265 (2000) 274; Y. Kodama, Massive WDM and TDM Soliton Transmission Systems, Kluwer Academic, Dordrecht, MA, ] R.Y. Hao, L. Li, Z.H. Li, W.R. Xue, G.S. Zhou, Opt. Commun. 236 (2004) ] R. Hirota, in: R.K. Bullough, P.S. Caudrey (Eds.), Direct Methods in Soliton Theory Solitons, Springer, Berlin, ] R. Hirota, Phys. Rev. Lett. 27 (97) 92; R. Hirota, J. Satsuma, J. Phys. Soc. Japan 40 (978) 6. 24]. Joshi, Phys. Lett. A 25 (987) 456; J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (983) ] A. Remani, B. Grammaticos, T. Bountis, Phys. Rep. 80 (989) ] R. Hirota, J. Math. Phys. 4 (973) ] V.. Serkin, A. Hasegawa, Phys. Rev. Lett. 85 (2000) ] R. Hirota, J. Satsuma, J. Phys. Soc. Japan 45 (978) 74; C. Rogers, W.F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, ew York, 982; J.J. immo, Phys. Lett. A 99 (983) 279.