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New Jacobi Elliptic Function Solutions for the Zakharov Equations

DOI: 10.1155/2012/854619

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A generalized -expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics. 1. Introduction In recent years, with the development of symbolic computation packages like Maple and Mathematica, searching for solutions of nonlinear differential equations directly has become more and more attractive [1–7]. This is because of the availability of computers symbolic system, which allows us to perform some complicated and tedious algebraic calculation and help us find new exact solutions of nonlinear differential equations. In 2008, Wang et al. [8] introduced a new direct method called the -expansion method to look for travelling wave solutions of nonlinear evolution equations (NLEEs). The method is based on the homogeneous balance principle and linear ordinary differential equation (LODE) theory. It is assumed that the traveling wave solutions can be expressed by a polynomial in , and that satisfies a second-order LODE . The degree of the polynomial can be determined by the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NPDEs. The coefficients of the polynomial can be obtained by solving a set of algebraic equations. Many literatures have shown that the -expansion method is very effective, and many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized -expansion method [9], the modified -expansion method [10], the extended -expansion method [11], the improved -expansion method [12], and the -expansion method [13] have been proposed. As we know, when using the direct method, the choice of an appropriate auxiliary LODE is of great importance. In this paper, by introducing a new auxiliary LODE of different literature [8], we propose the generalized -expansion method, which can be used to obtain travelling wave solutions of NLEEs. In our contribution, we will seek exact solutions of the Zakharov equations [14]: which are one of the classical models on governing the dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves, where is the perturbed number density of the ion (in the low-frequency response), is the slow variation amplitude of the electric field intensity, is the thermal transportation velocity of the


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