Several Constructive Methods In Solving Nonlinear Evolution Equations

This dissertation has mainly done the following two aspectsresearch: Firstly, with the aid of the symbolic computation softwareMaple, some methods for constructing the exact solutions of nonlin-ear evolution equations are presented and improved. Secondly, Dar-boux transformation for the BB equations has been studied. Thestructure of this paper is as follows: Chapter 1 is concerned withthe exposition of the development and the research situation of sev-eral subjects which will be discussed in this paper. Our main worksare presented at last. Based on the idea of solving nonlinear evolu-tion equations, algebraic method, algorithm reality, mechanization, inChapter 3, we firstly present the new extended first kind elliptic sub-equation method. Many interesting exact solutions of Konopelchenko-Dubrovsky equations are explored, including new bell profile solitarywave solutions, kink profile solitary wave solutions and triangular pe-riodic wave solutions. Secondly, new three Riccati equation methodis proposed to obtain solutions of di?erence-di?erential equations. Weexplore the discrete KdV equation and successfully obtain many newsolutions. At last, by using the generalized projected Riccati equationmethod and the hyperbolic auxiliary equation method, we obtain theexplicit solutions of the subsidiary ordinary di?erential equation. Thesolutions of a class of variable coe?cient equations(such as the generalKP and KdV equation with variable coe?cients) are derived by meansof a suitable transformation and the subsidiary ordinary di?erentialequation mentioned above. In chapter 3, we succesfully construct akind of new Darboux transformation for the BB equations. As an ap-plication, new soliton solutions and multi-soliton solutions of the BBequations are obtained. In chapter 4, we make a summary of the paperand give an outlook of the future research direction.