The Research Of Two Kinds Of Soliton Equations Based On Bilinear Method

Constructing the exact solutions of nonlinear evolution equations and seeking new in-tegrable coupled systems are two central problems in the study of nonlinear equations. In the recent years, the experts and scholars have developed many effective methods for solv-ing nonlinear soliton equations, for example, the Hirota direct method, the anti-scattering method, Painleve analysis method, Darboux transformation, Pfaffian technique, the classical and non classical Lie group method and so on. In this thesis, we study the exact solutions and coupled systems of many soliton equations with the help of the Hirota direct method and Pfaffian technique.The main results of the article are as follows:In the Chapter 1, the discovery and development of soliton are briefly reviewed. Then the situation of the Hirota direct method and Pfaffian technique are briefly introduced.In the Chapter 2, the definition of Pfaffian and the Pfaffian identities are ordinary introduced.In the Chapter 3, firstly, a variable-coefficient coupled Kadomtsev-Petviashvili(KP) system for the variable-coefficient KP equation is given out by employing the pfaffianization procedure. Secondly, Wronski-type pfaffian solution and the Gramm-type pfaffian solution of the coupled system are presented by Pfaffian identities.In the Chapter 4, firstly, Grammian solutions are generated for a variable-coefficient (2+1)-dimensional generalized breaking soliton equation by using the Jacobi identity for determinants and Grammian derivative formulae. Secondly, a Pfaffian extension for the equation is obtained through the Pfaffianization procedure; Third, the Wronski-type Pfaffian solution and Gramm-type Pfaffian solution of the resulting coupled system are also given.In the Last, the summary of this paper is given, and some problems which need to be further studied are pointed out.