New Traveling Wave Solutions to the Vakhnenko-Parkes Equation

We apply the improved expansion method to the Vakhnenko-Parkes equation. As a result, many new and more general exact solutions have been obtained for the equation. Comparing our solutions with those gained by other authors indicates that the improved expansion is more effective in solving the general solutions to differential equations. 1. Introduction Many phenomena in physics and other fields such as biology chemistry and mechanics. are described by nonlinear partial differential equations (NLPDEs). The investigation of traveling wave solutions to nonlinear partial differential equations (NLPDEs) plays an important role in the study of nonlinear physical phenomena. In the past several decades, both mathematicians and physicists have made significant progress in this direction. Many effective methods [1–10] have been presented such as exp-function method [1], Hirota’s method [2], variational iteration method [3], the homogeneous balance method [4], Backlund and Darboux transformation method [5], the sine-cosine function method [6], the Jacobi elliptic function method [7], and auxiliary equation method [8–10]. Recently, an interesting and important discovery has been made by Vakhnenko and Parkes [11], who have demonstrated that the reduced Ostrovsky equation [12] can be transformed to the new integrable equation In the literature, the traveling wave solutions of the Vakhnenko-Parkes equation (2) are investigated by the improved tanh function method introduced in [13, 14], auxiliary equation method [10], and -expansion method [15]. The present paper is motivated by the desire to improve the work made in [4, 10, 13–15] by proposing a new improved -expansion method to construct more general exact solutions of nonlinear partial differential equations (NLPDES). For illustration, we restrict our attention to the study of the Vakhnenko-Parkes equation (2) and successfully construct many new and more general exact solutions. The rest of this paper is organized as follows: we give the description of the improved -expansion method in Section 2. In Section 3, we apply this method to (2). In Section 4, some conclusions are given. 2. Description of the Improved -Expansion Method To make this paper entire, in here we enumerate the same method reported in [16]. Suppose that we have a NLPDE for in the form where is a polynomial in its arguments, which includes nonlinear terms and the highest order derivatives. The transformation , reduces (3) to the ordinary differential equation (ODE) By virtue of the extended tanh-function method, we assume that the solution of (4)