Lax Pair, Bäcklund Transformation and N-Soliton-Like Solution for a Variable-Coefficient Gardner Equation from Nonlinear Lattic

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector

J. Math. Anal. Appl. 336 (2007) 1443–1455 www.elsevier.com/locate/jmaa

Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefﬁcient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

Juan Li a,∗, Tao Xu a, Xiang-Hua Meng a, Ya-Xing Zhang a, Hai-Qiang Zhang a, Bo Tian a,b

a School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China b Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China Received 3 October 2006 Available online 27 March 2007 Submitted by M.C. Nucci

Abstract In this paper, a generalized variable-coefficient Gardner equation arising in nonlinear lattice, plasma physics and ocean dynamics is investigated. With symbolic computation, the Lax pair and Bäcklund transformation are explicitly obtained when the coefficient functions obey the Painlevé-integrable conditions. Meanwhile, under the constraint conditions, two transformations from such an equation either to the constant-coefficient Gardner or modified Korteweg–de Vries (mKdV) equation are proposed. Via the two transformations, the investigations on the variable-coefficient Gardner equation can be based on the constant-coefficient ones. The N-soliton-like solution is presented and discussed through the figures for some sample solutions. It is shown in the discussions that the variable-coefficient Gardner equation possesses the right- and left-travelling soliton-like waves, which involve abundant temporally-inhomogeneous features. © 2007 Elsevier Inc. All rights reserved.

Keywords: Soliton solutions; Integrable properties; Gardner equation; Symbolic computation

* Corresponding author. E-mail address: [email protected] (J. Li).

1. Introduction

The Korteweg–de Vries (KdV)-typed equations with the quadratic nonlinearity are impor- tant nonlinear models [1], which have been derived in many unrelated branches of sciences and engineering including the pulse-width modulation [2], mass transports in a chemical response theory [3], dust acoustic solitary structures in magnetized dusty plasmas [4] and nonlinear long dynamo waves observed in the Sun [5]. However, the high-order nonlinear terms must be taken into account in some complicated situations like at the critical density or in the vicinity of the critical velocity [6–8]. The modified KdV (mKdV)-typed equation, on the other hand, has re- cently been discovered, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in the supernova shells and Saturn’s F-ring [9]. If the quadratic and cubic nonlinear terms are both considered, the KdV equation becomes the Gardner equation, which is also called the combined KdV and mKdV (KdV–mKdV) equation [10–14]. The Gardner equation has been investigated in detail and can be used to model such physical situations as the dust-acoustic solitary waves in dusty plasmas [6], internal solitary waves in stable, stratified shear flows in ocean and atmosphere [15], ion acoustic waves in plasmas with a negative ion [16], interfacial solitary waves over slowly varying topographies [17] and wave motion in a nonlinear elastic structural element with large deflection [18]. In multifarious real physical backgrounds, the variable-coefficient nonlinear evolution equations (NLEEs) often can provide more powerful and realistic models than their constant- coefficient counterparts when the inhomogeneities of media and nonuniformities of boundaries are considered. Thereby, some inhomogeneous Gardner models with the time-dependent coefficients have been derived to describe a variety of interesting and significant phenomena in ocean dynamics, fluid mechanics and plasma physics, as follows:

• If the pycnocline lies midway between the sea bed and surface, the following variable- coefficient mKdV model [8,19,20], 2 ut + a(t)uux + κu ux + uxxx = 0, (1) where a(t) depends on the pycnocline location and κ is a constant, can describe the internal waves in a stratified ocean, as observed on the northwest shelf of Australia [21] and in the Gotland deep of the Baltic Sea [22]. • In an inhomogeneous two-layer shallow liquid, the governing equation modelling the long wave propagation is the extended KdV model with variable coefficients [23,24], 2 ut − 6α(t)uux − 6γu ux + θ(t)uxxx = 0, (2) where u(x, t) is proportional to the elevation of the interface between two layers, α(t) and θ(t) imply that the ratio of the depths of two layers may depend on the coordinate “t” [25]. • In the investigation on the dynamics hidden in the plasma sheath transition layer and inner sheath layer, a perturbed mKdV model is proposed as follows [26], 2 ψτ + 6ψψη + σψ ψη − ψηηη + h(τ)ψη = 0, (3) where σ is a constant, h(τ) is an analytic function and the physical meaning of ψ(η,τ) and h(τ) can be seen in Ref. [26].

In this paper, by virtue of symbolic computation [27], we will investigate the following generalized Gardner equation with the time-dependent coefficients, 2 ut + a(t)uux + b(t)u ux + c(t)uxxx + d(t)ux + f(t)u= 0, (4) J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1445 where u(x, t) is the amplitude of the relevant wave model, e.g., for the internal waves in a stratified ocean, x is the horizontal coordinate and t is the time, while the time-dependent coefficients a(t), b(t), c(t), d(t) and f(t) are all analytic functions and relevant to the background density and shear flow stratification. For a generalized variable-coefficient NLEE, it is not completely integrable unless the variable coefficients satisfy certain constraint conditions. It has been shown that in Refs. [28–30] the constraint conditions on the coefficient functions for some variable-coefficient NLEEs (e.g., the variable-coefficient KdV, nonlinear Schrödinger and Kadomatsev–Petviashvilli equations) to be mapped to the completely-integrable constant-coefficient counterparts are precisely the same as those for such equations to possess Painlevé properties. Thereby, we will first find the conditions for Eq. (4) to pass the Painlevé test, and then investigate its some integrable properties. The structure of this paper is as follows. In Section 2, through the Painlevé test, we derive the conditions for Eq. (4) to be Painlevé integrable, and then construct its Lax pair. In Section 3, using the Lax pair, we obtain the Bäcklund transformation and one-soliton-like solution of Eq. (4). In Section 4, under the Painlevé-integrable conditions, we provide two transformations from Eq. (4) either to the constant-coefficient Gardner or mKdV equation and the N-soliton-like solution of Eq. (4). Section 5 will be our discussions and conclusions.

2. Painlevé analysis and Lax pair for Eq. (4) with symbolic computation

In this section, to determine the constraint conditions for Eq. (4) to be Painlevé integrable, we will employ the Weiss–Tabor–Carnevale (WTC) procedure [31] to carry out the Painlevé analysis for Eq. (4). According to the WTC procedure, if the solutions of a given partial differential equation (PDE) are “single-valued” about the movable singularity manifolds, then this PDE has the Painlevé property. The generalized Laurent series expansion of u is of the form ∞ α j u = φ(x,t) uj (x, t)φ(x, t) , (5) j=0 where α is a negative integer, uj (x, t) and φ(x,t) are analytic functions in a neighborhood of the noncharacteristic singular manifold. 1/2 Through the leading order analysis, we obtain α =−1 and u0 =[−6c(t)/b(t)] φx . With the aid of symbolic computation, it is found that the resonances occur at j =−1, 3, 4, of which j =−1 corresponds to the arbitrariness of the singular manifold. On the other hand, the compatibility conditions at j = 3 and j = 4 are satisﬁed identically, if the variable coefﬁcients obey the following constraints, f(t)dt 2 f(t)dt a(t) = k1c(t)e ,b(t)= k2c(t)e , (6) where k1 and k2 are two arbitrary constants. Therefore, under constraints (6), we can say that Eq. (4) possesses the Painlevé property and might admit some notable properties like the Lax pair, Bäcklund transformation and N-soliton-like solution. In the following research, we will stay with constraints (6) and follow the Ablowitz–Kaup– Newell–Segur (AKNS) approach to construct the Lax pair of Eq. (4). The linear eigenvalue problems for Eq. (4) can be expressed as

Ψx = UΨ, Ψt = VΨ, (7) 1446 J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455

T where Ψ = (ψ1,ψ2) , T denotes the transpose of the matrix, while U and V are presented in the forms η β(x, t)u(x, t) A(x,t,η) B(x,t,η) U = , V = , (8) δ(x,t)v(x,t) −η C(x,t,η) −A(x, t, η) with f(t)dt 1 f(t)dt β(x,t) = e ,δ(x,t)= 1,v(x,t)=− k2e u + k1 , (9) 6 1 1 A =−4c(t)η3 − d(t)+ e f(t)dtc(t) k e f(t)dtu2 + k u η − k c(t)e f(t)dtu , 3 2 1 6 1 x (10) 1 1 B =−e f(t)dt 4c(t)uη2 + 2c(t)u η + k c(t)e2 f(t)dtu3 + k c(t)e f(t)dtu2 x 3 2 3 1

+ d(t)u+ c(t)uxx , (11)

2 − 1 1 C = e f(t)dt c(t) k e f(t)dt + k u η2 − k c(t)u η + k2c(t)e2 f(t)dtu3 3 1 2 3 2 x 18 2

1 1 1 1 1 + k k c(t)e f(t)dtu2 + k2c(t)u + k d(t)u+ k c(t)u + k d(t), (12) 9 2 1 18 1 6 2 6 2 xx 6 1 where η is a parameter independent of x and t. It is easy to prove that the compatibility condition Ut − Vx +[U, V]=0 gives rise to Eq. (4). The above-obtained Lax pair can assure the complete integrability of Eq. (4). In the next section, we will construct the Bäcklund transformation for Eq. (4) by virtue of the Lax pair.

3. Bäcklund transformation and one-soliton-like solution for Eq. (4)

By introducing a function Γ as ψ Γ = 1 , (13) ψ2 = = =−1 [ + ] Eqs. (7) with βx δx 0 and v 6 k2β(t)u k1 turn out to be the following Riccati-typed equations, 1 Γ = k β(t)u+ k Γ 2 + 2ηΓ + β(t)u, (14) x 6 2 1 2 Γt =−CΓ + 2AΓ + B. (15) Furthermore, we assume that Γ = G(η, Γ ), u˜ = u + F(η,Γ,Γx), (16) where Γ and Γ are a couple of different solutions of Eqs. (14) and (15), u and u˜ are two distinct solutions for Eq. (4), whereas G(η, Γ ) and F(η,Γ,Γx) are two analytic functions to be determined. Considering the invariance of Eqs. (14) and (15) with Γ and u˜, we can get

12η + k Γ 2Γ Γ =− 1 , u˜ = u − x , (17) k − 2k ηΓ + 1 2 1 2 β(t)(1 6 k2Γ ) J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1447 which provide us with 6 k β(t)(ω˜ − ω) Γ = tan 2 , (18) k2 6 2 where u =−ωx and u˜ =−˜ωx . Since expressions (9)–(12) are comparatively complicated in the format, then in the following, we will determine the Bäcklund transformation by virtue of software Mathematica. With the aid of symbolic computation, eliminating Γ in Eqs. (14) and (15) with the substitution of expressions (9)–(12) and (18) gives rise to the following Bäcklund transformation for Eq. (4), − f(t)dt k1 k1 k2 f(t)dt ωx +˜ωx = e − cos e (ω˜ − ω) k2 k2 6 6 k + 2η sin 2 e f(t)dt(ω˜ − ω) , (19) k2 6

2 k k − k − ω +˜ω = 1 c(t)ω − 4 1 η2c(t)e f(t)dt − 1 d(t)e f(t)dt − f (t)(ω +˜ω) t t 3k x k k 2 2 2 k k + 8η2c(t)ω − Q cos 2 e f(t)dt(ω˜ − ω) − R sin 2 e f(t)dt(ω˜ − ω) , x 6 6 (20) with 2 k 1 k − Q = 1 c(t)ω − k c(t)e f(t)dtω2 − 1 e f(t)dt 4c(t)η2 + d(t) − 4ηc(t)ω , 3k x 3 1 x k xx 2 2 6 − 2k R = 2 e f(t)dt 4c(t)η3 + d(t)η + 2 2 ηe f(t)dtc(t)ω2 k 3 x 2 2 − k1c(t)[2ωxη − ωxx]. 3k2 From expressions (19) and (20) by taking ω˜ = 0 as a vacuum solution, one can get the one- soliton-like solution of Eq. (4), written as 6 − ∂ H u =−2 e f(t)dt arctan tanh ηx − 4η3c(t) + ηd(t) dt , (21) k2 ∂x K where k2 2k η2 k k2 2k η2 k =±1,H= 1 + 2 + 1 ,K= 1 + 2 − 1 . (22) 36 3 6 36 3 6

4. Transformations for Eq. (4) with symbolic computation

In this section, via symbolic computation, we will construct two transformations from Eq. (4) either to the constant-coefﬁcient Gardner or mKdV equation when the Painlevé-integrable constraints hold. 1448 J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455

4.1. Transformation to the constant-coefﬁcient Gardner equation

Utilizing the method proposed in Refs. [29,30], Eq. (4) can be transformed into the following constant-coefﬁcient Gardner equation [14], 2 UT + μUUX + νU UX + UXXX = 0, (23) by the transformation as below: − μλ k − u = e f(t)dt − 1 + λe f(t)dtU X(x,t),T (t) , (24) 2ν 2k2 with k λ2 k λ2 k2 k μ2λ2 X(x,t) = 2 x + 2 1 − 2 c(t) − d(t) dt, 2 (25) ν ν 4k2 4ν k λ2 3/2 T(t)= 2 c(t)dt, (26) ν where k1, k2 = 0, λ = 0, μ and ν = 0 are all arbitrary constants.

4.2. Transformation to the constant-coefﬁcient mKdV equation

In addition, under the Painlevé-integrable conditions, we can also convert Eq. (4) to the constant-coefficient mKdV equation [6], 2 UT + 24U UX + UXXX = 0, (27) by the transformation − k − u = ρe f(t)dtU X(x,t),T (t) − 1 e f(t)dt, (28) 2k2 with 2 2 = 1 ρ k2 + ρ 2 − X(x,t) x k1c(t) 4k2d(t) dt, (29) 2 6 8 6ρ2k 2 (ρ2k )3/2 T(t)= √2 c(t)dt, (30) 48 6 where k1, k2 = 0 and ρ = 0 are all arbitrary constants. It is known that Eqs. (23) and (27) are both soliton equations and their initial value problems have been solved by the inverse scattering method [10]. Accordingly, the two equations possess such remarkable properties as the Lax pair, Bäcklund transformation, bilinear form, N-soliton solution, nonlinear superposition formula and an infinite number of conservation laws. Under constraints (6), Eq. (4) also admits the similar properties since transformations (24) and (28) can map this equation onto Eqs. (23) and (27), respectively. Additionally, some direct methods have given abundant analytic solutions of Eqs. (23) and (27), e.g., the new solitary wave solutions and Jacobi doubly periodic wave solutions [32]. Therefore, substituting those solutions into transformations (24) and (28) will give rise to the corresponding analytic solutions for Eq. (4), of which some may be different from those obtained by J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1449 other methods. Among the out-coming results, special attention will be paid to the N-soliton-like solution of Eq. (4). For example, using transformation (28), the one-soliton-like solution can be expressed as 6 − ∂ k − u = 2 e f(t)dt arctan eθ − 1 e f(t)dt k2 ∂x 2k2 pρ − k − = e f(t)dt sech(θ) − 1 e f(t)dt, (31) 2 2k2 where 2 3 2 2 ρ{24pk2x + 6p[k c(t) − 4k2d(t)] dt − p k ρ c(t)dt} θ = 1 √ 2 + ζ, 48 6k2 with p and ζ as two arbitrary constants. The two-soliton-like solution of Eq. (4) can be written as θ1 + θ2 = 6 − f(t)dt ∂ e e − k1 − f(t)dt u 2 e arctan θ +θ e , (32) k2 ∂x 1 + A12e 1 2 2k2 where { + [ 2 − ] − 3 2 2 } ρ 24pik2x 6pi k1c(t) 4k2d(t) dt pi k2ρ c(t)dt θi = √ + ζi, 48 6k2 2 2 pi and ζi (i = 1, 2) are arbitrary constants, and A12 =−(p1 − p2) /(p1 + p2) .Itisworth noting that there are abundant temporally-inhomogeneous features in expressions (31) and (32) under the influence of the time-dependent coefficient functions, as illustrated in Figs. 1–8. Furthermore, we can explicitly present the N-soliton-like solution of Eq. (4) in the sense of Ref. [6] as below, 6 − ∂ G(x, t) k − u = 2 e f(t)dt arctan − 1 e f(t)dt, (33) k2 ∂x F(x,t) 2k2 where [(N−1)/2] 2n+1 j=1 θi G(x, t) = g(i1,i2,...,i2n+1)e j , n=0 N C2n+1 [N/2] 2n j=1 θi F(x,t)= g(i1,i2,...,i2n)e j , = n 0 N C2n (p −p )2 (n) − ik il 2 , for n 2, g(i1,i2,...,in) = k

It is worth noting that Eq. (23) can also be transformed into Eq. (27), since the former is a special case of Eq. (4) with a(t) = μ, b(t) = ν, c(t) = 1 and d(t) = f(t)= 0. So, under the Painlevé-integrable conditions, we can also get the N-soliton-like solution of Eq. (4) via transformation (24). As shown in expression (33), those solutions imply abundant temporally- inhomogeneous features, which may be used to depict some interesting physical phenomena in nonlinear lattice, plasma physics and ocean dynamics.

5. Discussions and conclusions

In this paper, based on the computerized symbolic computation, the variable-coefficient Gard- ner equation arising from nonlinear lattice, plasma physics and ocean dynamics, has been investigated. The Painlevé analysis has been carried out to determine the constraints for this equation to be Painlevé integrable. Under the constraint conditions, we have not only derived the Lax pair and Bäcklund transformation for Eq. (4), but also constructed two transformations from such an equation either to the constant-coefficient Gardner or mKdV equation. Additionally, by one of the transformations, the analytic N-soliton-like solution for Eq. (4) has been explicitly obtained. When the Painlevé-integrable conditions are satisfied, the variable-coefficient Gardner equation admits many other properties, e.g., the bilinear form, an infinite number of conservation laws and Darboux transformation. Among the above solutions to Eq. (4), we would like to concentrate ourselves on expressions (31) and (32), which are the one- and two-soliton-like solutions and can be used to model the wave propagations in nonlinear lattice, plasma physics and ocean dynamics. In order to re- veal the physical properties, the actual application of the soliton-like solution often requires an advisable value for the involved parameters and functions. Hence for the picture drawing and qualitative analysis, we choose some values for those parameters and functions in line with the nonuniform backgrounds of the variable-coefficient Gardner model under investigation. Figure 1, with the constant values for the variable parameters and functions, provides us with a special travelling wave via expression (31). When the coefficient functions are chosen as noncon- stant values, the nontravelling effects will appear in the propagation process of solitary waves. In Fig. 2, by choosing c(t) = 1 + 0.2t, it is shown that the velocity of solitary wave is variable along the propagation direction. Correspondingly, the effect of coefficient function f(t)is presented in Fig. 3, which indicates that the amplitude and equilibrium position are both periodically variable with the evolution of time, but the velocity is invariant.

Fig. 1. The one-soliton solution surface for expression (31) is the special travelling wave case, with p = 1, k1 = 1, k2 = 2, ρ = 2, ζ = 1, c(t) = 1andd(t) = f(t)= 0. J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1451

Fig. 2. The one-soliton-like solution surface for expression (31) to be compared with in Fig. 1 is beyond the travelling wave, with the same parameters and functions as in Fig. 1 except that c(t) = 1 + 0.2t.

Fig. 3. The one-soliton-like solution surface for expression (31) to be compared with Fig. 1 shows the effect of f(t), with the same parameters and functions as in Fig. 1 except that f(t)= 0.3sin(0.75t).

Fig. 4. The head-on collision of two travelling waves via expression (32) with p1 = 1.5, p2 = 3.2, k1 =−1, k2 = 1, ρ = 1, ζ1 = ζ2 = 0, c(t) = 1.5andd(t) = f(t)= 0.

Figure 4 depicts the head-on collision of one left-going soliton and one right-going soliton for expression (32). Figure 5 presents a set of three photographs for Fig. 4 taken at three different times, which reﬂect the elastic collision of two solitary waves along opposite directions of propagation. For comparison with Fig. 4, by choosing f(t)= 0.5sin(t), we can see the head-on collision of two nontravelling waves in Fig. 6, which shows that the amplitudes and equilibrium position are both periodically variable with the wave propagation, while the velocities are invariant. In like manner, we can also draw a set of photographs for Fig. 6 to display the bidirectional collision process of two nontravelling waves. In contrast, Fig. 7 shows the collision of two solitary waves along same directions of propagation for expression (32). Through a set of 1452 J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455

(a) (b)

(c)

Fig. 5. The set of three photographs for Fig. 4 to present the head-on collision of two solitary waves at (a) t =−100, (b) t = 0and(c)t = 100.

Fig. 6. The head-on collision of two nontravelling waves via expression (32) to be compared with Fig. 4, with the same parameters and functions as in Fig. 4 except that k1 =−6, k2 = 5andf(t)= 0.5sin(t). photographs for Fig. 7 taken at an equal temporal interval, Fig. 8 demonstrates that the large- amplitude solitary wave with faster velocity overtakes the small-amplitude one, after collision, the shorter is left behind. From the above discussions, it can be concluded that Eq. (4) admits bidirectional wave interactions including head-on and overtaking collisions [33,34], unlike the KdV equation which only exhibits the unidirectional interactions. Since the problem of bidirectional solitary waves has been reported in a uniform layer of water [35], it is expected that the bidirectional soliton- like solutions to Eq. (4) may be used to describe such interesting physical phenomena with the consideration of the inhomogeneities of media and nonuniformities of boundaries in nonlinear lattice, plasma physics and ocean dynamics. For many other variable-coefﬁcient NLEEs, we can also determine their Painlevé-integrable conditions ﬁrstly, and then make further investigations on their integrable properties. Further- J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1453

Fig. 7. The overtaking collision of two travelling waves via expression (32) with p1 = 4, p2 = 8, k1 =−1, k2 = 2.5, ρ = 1, ζ1 = ζ2 = 0, c(t) = 0.5andd(t) = f(t)= 0.

(a) (b)

(c)

Fig. 8. The set of three photographs for Fig. 7 to present the overtaking collision of two solitary waves at three different times (a) t =−8, (b) t = 0and(c)t = 8. more, when the constraint conditions hold, it is practicable to construct the transformations to their constant-coefficient counterparts. Through converting the variable-coefficient NLEEs to their constant-coefficient counterparts, the investigations on those variable-coefficient equations can be based on the conventional ones.

Acknowledgments

We thank the referee for his/her helpful suggestions. We thank Professor Y.T. Gao for his valuable comments and discussions. We also express our thanks to each member of our discussion group for their suggestions, especially Mr. X. Lü. This work has been supported by the Key Project of Chinese Ministry of Education (No. 106033), by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060006024), Chinese Ministry of Education, and by the National Natural Science Foundation of China under Grant No. 60372095. 1454 J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455

References

[1] B. Tian, Y.T. Gao, Phys. Plasmas 12 (2005) 054701; 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, Eur. Phys. J. D 33 (2005) 59; B. Tian, Y.T. Gao, Phys. Plasmas (Lett.) 12 (2005) 070703; Y.T. Gao, B. Tian, Phys. Lett. A 349 (2006) 314. [2] D. Chudnovsky, G. Chudnovsky, Proc. Natl. Acad. Sci. USA 96 (1999) 12263. [3] Z. Horii, Phys. Lett. A 306 (2002) 45; Z. Horii, Phys. A 337 (2004) 398. [4] L.P. Zhang, J.K. Xue, Chaos Solitons Fractals 23 (2005) 543. [5] J. Mason, E. Knobloch, Phys. D 205 (2005) 100. [6] A.H. Khater, A.A. Abdallah, O.H. El-Kalaawy, D.K. Callebaut, Phys. Plasmas 6 (1999) 4542; R. Hirota, J. Phys. Soc. Japan 33 (1972) 1456. [7] Y. Nejoh, Phys. Fluids B 4 (1992) 2830. [8] R. Grimshaw, E. Pelinovsky, T. Talipova, Phys. D 132 (1999) 40. [9] Y.T. Gao, B. Tian, Phys. Plasmas (Lett.) 13 (2006) 120703. [10] M. Wadati, J. Phys. Soc. Japan 38 (1975) 673; M. Wadati, J. Phys. Soc. Japan 38 (1975) 681. [11] M.N.B. Mohamad, Math. Methods Appl. Sci. 15 (1992) 73. [12] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, A. Slunyaev, Chaos 12 (2002) 1070. [13] G.F. Yu, H.W. Tam, J. Math. Anal. Appl. 330 (2007) 989. [14] A.V. Slyunyaev, J. Exp. Theor. Phys. 92 (2001) 529; K.W. Chow, R.H.J. Grimshaw, E. Ding, Wave Motion 43 (2005) 158. [15] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, T. Talipova, Phys. D 159 (2001) 35. [16] S. Watanabe, J. Phys. Soc. Japan 53 (1984) 950. [17] K.R. Helfrich, W.K. Melville, J.W. Miles, J. Fluid Mech. 149 (1984) 305; K.R. Helfrich, W.K. Melville, J. Fluid Mech. 167 (1986) 285. [18] Q. Han, L.M. Dai, M.Z. Dong, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 607. [19] V. Djordjevic, L. Redekopp, J. Phys. Oceanogr. 8 (1978) 1016. [20] T. Kakutani, N. Yamasaki, J. Phys. Soc. Japan 45 (1978) 674; J.W. Miles, Tellus 31 (1979) 456; J.W. Miles, Tellus 33 (1981) 397. [21] P.E. Holloway, E. Pelinovsky, T. Talipova, B. Barnes, J. Phys. Oceanogr. 27 (1997) 871. [22] T. Talipova, E. Pelinovsky, T. Kouts, Oceanology 38 (1998) 33. [23] J.A. Gear, R. Grimshaw, Phys. Fluids 26 (1983) 14; K.G. Lamb, L. Yan, J. Phys. Oceanogr. 26 (1996) 2712. [24] B.A. Malomed, V.I. Shrira, Phys. D 53 (1991) 1. [25] W.P. Hong, M.S. Yoon, Z. Naturforsch. A 56 (2001) 366. [26] H.L. Liu, M. Slemrod, Appl. Math. Lett. 17 (2004) 401. [27] M.P. Barnett, J.F. Capitani, J. Von Zur Gathen, J. Gerhard, Int. J. Quantum Chem. 100 (2004) 80; G. Das, J. Sarma, Phys. Plasmas 6 (1999) 4394; Z.Y. Yan, H.Q. Zhang, J. Phys. A 34 (2001) 1785; Sirendaoreji, J. Phys. A 32 (1999) 6897; R. Ibrahim, Chaos Solitons Fractals 16 (2003) 675; W.P. Hong, S.H. Park, Int. J. Mod. Phys. C 15 (2004) 363; F.D. Xie, X.S. Gao, Commun. Theor. Phys. 41 (2004) 353; B. Li, Y. Chen, H.N. Xuan, H.Q. Zhang, Appl. Math. Comput. 152 (2004) 581; B. Tian, Y.T. Gao, Phys. Lett. A 342 (2005) 228; B. Tian, Y.T. Gao, Phys. Lett. A 359 (2006) 241; Y.T. Gao, B. Tian, C.Y. Zhang, Acta Mech. 182 (2006) 17; B. Tian, H. Li, Y.T. Gao, Z. Angew. Math. Phys. 56 (2005) 783. [28] N. Joshi, Phys. Lett. A 125 (1987) 456; L. Hlavatý, Phys. Lett. A 128 (1988) 335; P.A. Clarkson, IMA J. Appl. Math. 44 (1990) 27; J.P. Gazeau, P. Winternitz, J. Math. Phys. 33 (1992) 4087. J. Li et al. / J. Math. Anal. Appl. 336 (2007) 1443–1455 1455

[29] B. Tian, W.R. Shan, C.Y. Zhang, G.M. Wei, Y.T. Gao, Eur. Phys. J. B (Rapid Not.) 47 (2005) 329. [30] B. Tian, G.M. Wei, C.Y. Zhang, W.R. Shan, Y.T. Gao, Phys. Lett. A 356 (2006) 8. [31] J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (1983) 522. [32] J.F. Zhang, Int. J. Theor. Phys. 37 (1998) 1541; J. Yu, Math. Methods Appl. Sci. 23 (2000) 1667; W.P. Hong, Nuovo Cimento Soc. Ital. Fis. B 115 (2000) 117; Z.T. Fu, S.D. Liu, S.K. Liu, Chaos Solitons Fractals 20 (2004) 301; E. Yomba, Phys. Lett. A 336 (2005) 463; X.Q. Zhao, H.Y. Zhi, Y.X. Yu, H.Q. Zhang, Appl. Math. Comput. 172 (2006) 24. [33] Y.S. Li, J.E. Zhang, Chaos Solitons Fractals 16 (2003) 271. [34] J.E. Zhang, Y.S. Li, Phys. Rev. E 67 (2003) 016306. [35] Y.S. Li, W.X. Ma, J.E. Zhang, Phys. Lett. A 275 (2000) 60; J.M. Dye, A. Parker, J. Math. Phys. 42 (2001) 2567; Y.S. Li, J.E. Zhang, Phys. Lett. A 284 (2001) 253.