Exact Solutions Of Nonlinear Evolution Equation And Formulation Of The Integrable Systems As Well As Its Expanding Integrable System

The major contents in this paper include:the exact solutions of nonlinear evolution equation and the formulation of the integrable system as well as its expanding integrable system. In the first chapter, the development of soliton theory and integrable system together with its research meaning are summarized, at the same time, the methods of searching exact solutions of nonlinear evolution equation are presented. In the second chapter, firstly, Painleve property of the modified C-KdV equation are introduced. Secondly, with the help of Maple software, the exact soliton solutions of the modified C-KdV equation are obtained by applying the standard truncated expansion method and the extended truncated expansion method respectively. The auto-backland transformation of equation may be obtained as well. The third chapter can be divided into three parts. In the first part, the general theory and method of integrable system are introduced. In the second part, new Lie algebra and its related two kinds of loop algebras are constructed, from which two new isospectral problems are designed. Using zero curvature equation, two Liouville integrable system are obtained which posses Hamilton structure and bi-Hamilton structure respectively, and the first system can be reduced to generalized Schrodinger hierarchy. Finally, the super Lie algebra is constructed, by using of the designed isospectral problem and the supertrace identity, the super-Burgers soliton hierarchy as well as its super Hamilton are obtained.