New Self-Backlund Transformation And Exact Solutions Of Two Nonlinear Equations

With the rapid development of nonlinear science, finding the exact solutions of the nonlinear evolution equations (NLEES) becomes an important topic in mathematics and physics. At present, a vast variety of direct methods finding the exact solutions of NLEES have been presented including the homogeneous balance method、Backlund transformation method、Hirota bilinear method and so on. As a direct method, the homogeneous balance method is very effective for finding special exact solutions of NLEES. Special exact solutions of many nonlinear equations in mathematical physics have been obtained by using it. In recent years, there are the extension of the homogeneous balance method and the modified homogeneous balance method.This paper obtains the new auto-Backlund transformations and exact solutions of two nonlinear evolution equations by using the modified homogeneous balance method that is based on the theories of the homogeneous balance method and Hirota bilinear method. This paper consists of the following three chapters.In Chapter1, we briefly introduce the background and history about the soliton theory and view some basic methods seeking exact solutions for the nonlinear evolution equations as well as the general steps by using the homogeneous balance method and the modified homogeneous balance method. Then, the major work of this paper is presented.In Chapter2, firstly we obtain a new auto-Backlund transformation of the generalized Boussinesq equation by using the modified homogeneous balance method. The auto-Backlund transformation only involves one quadratic homogeneity equation written as a bilinear equation. Based on the auto-Backlund transformation, the two-soliton solution of the generalized Boussinesq equation is obtained by using ε-expansion method. Then, a simplified version of the modified homogeneous balance method is given and the auto-Backlund transformation of the generalized Boussinesq equation is re-derived.In Chapter3, a new auto-Backlund transformation and four groups exact solutions of the Boiti-Leon-Pempinelli (BLP) equations are presented by using the modified homogeneous balance method.