Non-linear Differential Difference Equations And Its Louville Integrability Of Integrable Coupling System

The study of soliton theory and integrable coupling system has been developed, and there are many problems in many scientific fields. The integrable coupling system is found in the study of the non centered Virasoro symmetric algebra or integrable system. Many methods have been found for integrable coupling: perturbation method; the method of expanding the corresponding Lax pair; extending the method of the new loop algebra; using the method of semi direct sum of Lie algebras, etc..This paper is composed of six chapters, which mainly study the integrability and integrable coupling system of nonlinear differential difference equations, and discuss the Hamilton structure and integrability of integrable systems.In the first chapter , we introduce the origin and development of soliton, the application and significance of soliton theory.In the second chapter, we introduce a new family of integrable differential difference equations derived from the discrete zero curvature equation.In the third chapter, prove that the new differential - liouville integrability of differential equations.In the fourth chapter, a three-order spectrum problem and corresponding differential equation.The fifth chapter, Liouville integrability in differential equations .The sixth chapter is the summary and the prospect of this paper.