Research On Integrability For Several Kinds Of Soliton Equations

In this paper, integrability problems of some important soliton equations, which include: the exact solution, B¨acklund transformation, Lie symmetry, conservation law and so on, are studied using Bell polynomial method, the Riemann theta function periodic wave solutions method, Lie symmetry analysis method.In the first chapter, the research background of soliton theory is introduced, then we introduce the research background and current situation of three kinds of methods for investigating integrability of soliton equations. Finally, the topic of this article and the main work are briefly introduced.In the second chapter, Bell-polynomial method is used to study the integrability of a(3+1)-dimensional nonlinear evolution equation to get the bilinear representation and B¨acklund transformation of the equation. Then, the exact solutions are obtained by means of linear superposition principle and homoclinic test approach. The solution contains -wave solution, periodic solitary wave solution, and some image simulation examples of the solutions are given.The third chapter is devoted to devising a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota-Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. The one-periodic and two-periodic wave solutions of the equation are obtained. The relations between the periodic wave solutions and soliton solutions are rigorously established.In the fourth chapter, Lie symmetry analysis is performed on the DrinfeldSokolov-Wilson system in order to get the corresponding Lie algebra and similarity reductions of the system. In addition, we construct conservation laws for one special form of the system using the elegant Noether approach, and construct conservation laws for the system via the new conservation theorem. Thus the system is integrable in sense of conservation laws.In the fifth chapter, some conclusions and prospects are presented.