Some Studies On Some Types Of Expanding Model Of The Integrable Systems And Painlevé Analysis Of Non-linear Evolution Equation

The study contents in this paper mainly include:the expanding models of the integrable system and the Painleve analysis of the non-linear evolution equation. In the first chapter, history of soliton study and integrable system are presented. The second chapter divided into three parts:In the first part, firstly, based on a new algebra B2, a isospectral problem is designed. Using generalized zero curvature equation, a Lax integrable system is obtained. Nest, two types of expanding Lie algebras G1 and G2 based on Lie algebra B2 are constructed, two types of expanding models of above-mentioned Lax integrable system are obtained by designing two different isospectral problems. In the second part, based on the idea of semidirect sum of Lie algebra and two isospectral problems are designed. Two kinds of Liouville integrable systems are obtained which poss Hamilton structure by use of variational identity, and can be reduced to expanding integrable models of AKNS integrable system. In the third part, expand loop algebras B2 and B3, two types of Liouville integrable systems are obtained which poss Hamilton structure and the Bi-Hamilton structure by using Generalized Tu Formula, respectively. All of them can be reduced to S-mKdV equation hierarchy. In the third chapter, firstly, check the Painleve integrablity for the Burgers-KdV equation by use of the ARS algorithm and WTC-Kruskal algorithm, respectively. Secondly, two types of special exact solutions of the Burgers-KdV equation are obtained by the standard truncation expansion and Extend Painleve expansion and the figure of parts of the solutions are presented by Maple program.