Elsevier

Physics Letters A

Volume 369, Issues 1–2, 10 September 2007, Pages 62-69
Physics Letters A

Application of Exp-function method for nonlinear evolution equations with variable coefficients

https://doi.org/10.1016/j.physleta.2007.04.075Get rights and content

Abstract

In this Letter, the Exp-function method with the aid of symbolic computational system Maple is used to obtain generalized solitary solutions and periodic solutions of a generalized Zakharov–Kuznetsov equation with variable coefficients. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.

Introduction

The investigation of exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. The importance of obtaining the exact solutions, if available, of those nonlinear equations facilitates the verification of numerical solvers and aids in the stability analysis of solutions.

Many effective methods [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23] have been presented such as variational iteration method [2], [6], [12], [23], Adomian decomposition method [18], homotopy perturbation method [3], F-expansion method [5], [22], and others. A complete review on the field are available on [4].

Very recently, He and Wu [14] proposed a straightforward and concise method, called Exp-function method, to obtain generalized solitary solutions and periodic solutions, applications of the method can be found in [15], [16], [17] for solving nonlinear evolution equations arising in mathematical physics. The solution procedure of this method, with the aid of Maple, is of utter simplicity and this method can easily extended to other kinds of nonlinear evolution equations.

The present paper is motivated by the desire to extend the Exp-function method to a generalized Zakharov–Kuznetsov equation with variable coefficients, which reads [24]ut+α(t)uux+β(t)u2ux+uxxx+γ(t)uxyy=0, where α(t), β(t) and γ(t) are arbitrary function of t. Eq. (1) includes considerable interesting equations, such as KdV equation, mKdV equation, ZK equation and mZK equation. For the special case α(t)=1, β(t)=0 and γ(t)=k, Moussa derived the similarity reductions and some explicit solutions of Eq. (1) in [25]. Exact solutions with solitons and periodic structures of Eq. (1) with α(t)=a, β(t)=0, γ(t)=1 were obtained by sinecosine algorithm in [26].

Section snippets

Exact solution of the generalized Zakharov–Kuznetsov equation with variable coefficients

Using the transformationu=U(η),η=kx+ly+τ(t)dt, where k, l are constants, τ(t) is an integrable function of t to be determined later, Eq. (1) becomesτ(t)u+α(t)kuu+β(t)ku2u+[k3+γ(t)kl2]u=0, where the prime denotes the differential with respect to η.

In view of the Exp-function method, we assume that the solution of Eq. (3) can be expressed in the formu(η)=n=cdanexp(nη)m=pqbmexp(mη), where c, d, p and q are positive integers which are unknown to be determined later, an and bm are unknown

Conclusion

The Exp-function method with a computerized symbolic computation system Maple has been successfully used to obtain generalized solitary solutions and periodic solutions of a generalized Zakharov–Kuznetsov equation with variable coefficients.

Finally, it is worthwhile to mention that the Exp-function method can be also extended to other nonlinear evolution equations with variable coefficients, such as the mKdV equation [27], the (3+1)-dimensional Burgers equation [28] and so on. The Exp-function

Acknowledgements

The authors would like to express their great thankful to the referees for their useful comments and discussions.

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