Analytic N-solitary-wave solution of a variable-coefficient Gardner equation from fluid dynamics and plasma physics

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Abstract

In this paper, the investigation is focused on a variable-coefficient Gardner equation with quadric and cubic nonlinearities from fluid dynamics and plasma physics. Using the Hirota bilinear method, the one-, two- and three-solitary-wave solutions of the variable-coefficient Gardner equation are derived, and the analytic N-solitary-wave solution is presented for the first time in this paper with the aid of symbolic computation. Figures are plotted to illustrate the solutions obtained in this paper.

Introduction

In the nonlinear science, many important phenomena in various fields can be described by the nonlinear evolution equations (NLEEs) [1], [2]. Searching for exact soliton solutions of NLEEs plays an important and significant role in the study on the dynamics of those phenomena. Up to now, many effective methods have been presented, such as the inverse scattering transformation [3], the Bäcklund transformation [4], the Darboux transformation [5], the homogeneous balance method [6], the Hirota bilinear method [7] and the Lie group method [8]. To our knowledge, most of the aforementioned methods are related to the constant-coefficient models. Recently, much attention has been paid to the variable-coefficient nonlinear equations which can describe many nonlinear phenomena more realistically than their constant-coefficient ones [9]. The Gardner equation, or the extended Korteweg–de Vries (KdV) equation can describe various interesting physics phenomena, such as the internal waves in a stratified ocean [10], the long wave propagation in an inhomogeneous two-layer shallow liquid [11] and ion acoustic waves in plasma with a negative ion [12]. In this paper, we investigate a variable-coefficient Gardner equation [13],ut+f(t)uxxx+g(t)uux+h(t)u2ux+γ(t)ux+τ(t)u=0,where the amplitude u(x,t) is the function of spatial variable x and time variable t, (x,t)R2. The coefficients f(t), g(t), h(t), γ(t) and τ(t) are differentiable functions of t. Generally speaking, Eq. (1) is not completely integrable in the sense of the inverse scattering scheme. It contains some important special cases:

  • 1.

    When h(t)=0, γ(t)=0 and τ(t)=0, Eq. (1) reduces to the variable-coefficient KdV equation,ut+f(t)uxxx+g(t)uux=0.Especially whenf(t)=g(t)a+bf(t)dt,Eq. (2) possesses the Painlevé property [14], [15]. The Bäcklund transformation, Lax pair, similarity reduction and special analytic solutions of Eq. (2) have been obtained [16], [17], [18], [19], [20].

  • 2.

    When g(t)=-6a(t), h(t)=-6r, γ(t)=0 and τ(t)=0, Eq. (1) reduces to the following equation [11], [21], [22], [23]ut-6a(t)uux-6ru2ux+f(t)uxxx=0,which describes strong and weak interactions of different mode internal solitary waves, etc.

  • 3.

    When f(t)=r, g(t)=6α, h(t)=6β, γ(t)=0 and τ(t)=0, Eq. (1) becomes the constant-coefficient Gardner equation [24], [25]:ut+6αuux+6βu2ux+ruxxx=0,with r, α, and β as constants. It is widely applied to physics and quantum fields, such as solid-state physics, plasma physics, fluid dynamics and quantum field theory [26], [27], [28].

  • 4.

    When g(t)=6, f(t)=1, h(t)=0 and γ(t)=0, Eq. (1) reduces to the following constant-coefficient KdV equation with perturbed termut+6uux+uxxx+τ(t)u=0,which possesses the Painlevé property. If τ(t)=0 or τ(t)=12(t-t0), it corresponds to the well-known standard [29], [30] and the cylindrical KdV equations [31], [32], respectively.

The Hirota bilinear method [33], [34] provides a powerful tool to find exact solutions of several NLEEs. Through a dependent variable transformation, the NLEE can be written in bilinear form via which the N-soliton solution can be obtained in the form of an Nth-order polynomial in N exponential. In this paper, through a modified dependent variable transformation, the variable-coefficient Gardner equation is transformed into its bilinear form under certain coefficient constraint. And then with the help of symbolic computation [35], [36], we derive the multi-solitary-wave solutions of Eq. (1) under certain coefficient constraints, and the analytic N-solitary-wave solution is presented.

This paper is arranged as follows: in Section 2, through a modified dependent variable transformation, the variable-coefficient Gardner equation is transformed into the bilinear form under certain coefficient condition. In Section 3, with the aid of symbolic computation, the one-, two- and three-solitary-wave solutions are obtained through the formal parameter expansion technique, and the N-solitary-wave solution in explicit form is also presented. Section 4 is devoted to the conclusions and discussions on the results in this paper.

Section snippets

Bilinear form

In this section, we will transform Eq. (1) into its bilinear form through a modified dependent variable transformation. Supposingu(x,t)=k(t)xw(x,t),in Eq. (1) and integrating once with respect to x yield the following equation,k(t)w+k(t)wt+f(t)k(t)wxxx+12g(t)k(t)2wx2+13h(t)k(t)3wx3+r(t)k(t)wx+τ(t)k(t)w=0,with the integration constant as zero. Then introducing the transformationw=arctanG(x,t)F(x,t),where F(x,t) and G(x,t) are differentiable functions of x and t into Eq. (5) yields[k(t)+τ(t)k(

Multi-solitary-wave solutions

In this section, we will obtain the one-, two-, and three-solitary-wave solutions of Eq. (1) by the formal parameter expansion technique based on its bilinear form (10), (11) with the help of symbolic computation, and the N-solitary-wave solution in explicit form is also presented.

Expand F(x,t) and G(x,t) as the integral power of ε,F(x,t)=1+i=1εiFi(x,t),G(x,t)=i=1εiGi(x,t).Notingψ=Dt+f(t)Dx3+r(t)Dx,in Eq. (10), substituting expressions (19), (20) into the bilinear form and equating the like

Conclusions and discussions

The Gardner equation can be used to describe several important physical phenomena, specially those in the fluid dynamics and plasma physics. The variable-coefficient Gardner equation, i.e., Eq. (1) can describe those nonlinear physical mechanism more powerfully and realistically than the constant-coefficient one. Generally speaking, Eq. (1) is not completely integrable in the sense of inverse scattering scheme. In this paper, the Hirota bilinear method has been successfully used to investigate

Acknowledgements

We express our sincere thanks to the Editor, Referees, Prof. B. Tian and other members of our discussion group for their valuable comments. This work has been supported by the Beijing Excellent Talent Fund (No. 60624001), by the National Natural Science Foundation of China under Grant Nos. 60772023 and 60372095, by the Key Project of Chinese Ministry of Education (No. 106033), by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001, Beijing

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