Application of Exp-function method for nonlinear evolution equations with variable coefficients
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] where , and 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 , and , 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 , , were obtained by sinecosine algorithm in [26].
Section snippets
Exact solution of the generalized Zakharov–Kuznetsov equation with variable coefficients
Using the transformation where k, l are constants, is an integrable function of t to be determined later, Eq. (1) becomes 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 form where c, d, p and q are positive integers which are unknown to be determined later, and 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 ()-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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