Study On Exact Solutions For The (2+1) Dimensional Breaking Soliton Equations And WBK Equtions

In recent years, non-linear science has a rapid development as a new subject. The soliton theory as an important branch of the Nonlinear Science has a great development since the 1960s, and it has been well studied and widely applied, in fluid mechanics, plasma, optics, communications and other fields of natural science. Searching for the exact solutions of the nonlinear evolution equations (NLEEs) has become the key point of the study on non-linear science, and also the difficulty one. Up to now, there are many kinds of methods to obtain the exact solutions of the NLEEs such as Inverse Scattering method, Hirota method, Wronskian technique, Traveling-wave method.Based on the theory of the NLEEs, this paper studies several kinds of significant methods, and we obtain the exact solutions of the (2+1) dimensional breaking soliton equations and WBK Equation.The structure of the present paper is organized as follows: In chapter one, we first introduce the history and development of the soliton theory, and then by examples we explain several commonly used methods for solving the NLEEs.In chapter two, we study the Hirota method. It is a direct method for the exact solutions of the NLEEs, which developed by Hirota in the 1970s. We introduce the bilinear operator, the special properties of the bilinear operator and the commonly used three kinds of linear transformations. Then we explain the detailed process of the Hirota method by solving the (2+1) dimensional breaking soliton equations.The third chapter describes the Wronskian technique. The Wronskian technique construct a Wronskian determinant by using the bilinear form of the NLEEs, and then substitute it into the bilinear equation to verify the correctness. We use the Wronskian technique to obtain the Wronskian form N-soliton solution of the (2+1)-dimensional breaking soliton equations, and give the corresponding proof.In chapter four we introduce the Traveling-wave method. By solving the Whitham-Broer-Kaup (WBK) equations, we explain the specific solution process of the Tanh-function method and the Exp-function respectively. By comparing the solutions, we confirm that the solutions obtained by using the Exp-function method are more generalized than those obtained by the Tanh-function method.