Some Approaches For Generating Exact Solutions To Soliton Equations And Applications Of Darboux And Backlund Transformations

In this dissertation, we mainly consider the problem of some methods for constructing the solutions of nonlinear evolution equations in the theory of solitons, proposed a series of methods of sovling the nonlinear evolution equations.The paper consists of the following chapters:Chapter 1 of this dissertation is devoted to reviewing the history and development of soliton theory, some methods for finding exact solutions of nonlinear evolution equations, with an emphasis on some achievements on the subjects at home and abroad.In chapter 2,based on the idea of algebraic method, algorithm reality and mechanization for solving nonlinear evolution equation. By using of Wu method and symbolic computation, we present two kinds of methods for obtaining the exact solutions of nonlinear evolution equations: the one is the generalized auxiliary equation method. The formal solution is assumed as finite logarithm Laurent expansion, where the auxiliary is the second ordinary differential equation with constant coefficients. By employing the homogeneous balance method and the characteristic polynomial of ordinary differential equation, we have obtained the exact solutions to mix KdV-MKdV equation and (2+1)-dimensional KP equation. The other is based on the Riccati equation, we introduce Backlund transformations for the standard Riccati equation with generate new exact solutions by using its simple and know solutions. Finally, we work out the traveling-wave solutions of the (2+1)-dimension Toda lattice equation, the discrete modified KdV equation, respectively.Chapter 3 is mainly focused on Backlundtransformations and Darboux transformations. The first section is to introduce the application of the Backlund transformation. By using of the zero curvature equation, the generalized mKdV equation, Liouville equation and Sine-Gordan equation, Sinh-Gordan equation are generated via polynomial expansions. As well as we investigate a kind of formal B acklundtransformation for the generalized Sine-Gordan equation. The rest of the chapter is to present Darboux transformation, with the study of normal transformation of the eigenvalue problem, the necessary and sufficient condition for the eigenvalue problem (3.2.1) to become the general eigenvalue problem (3.2.2) through normal transformation containing noηis pointed out and the expression of normal transformation and the function u, vare given. Then further explained that the nonlinear evolution equations corresponding to eigenvalue problem (3.2.2)can be transformed into nonlinear equations corresponding to eigennvalue problem (3.2.1).