Several Soliton Equations And Their Coupling Families

Searching for a new integrable system and its expansion is one of the most important topics in the theory of integrable systems.This paper mainly includes the following aspects:Based on a new Loop algebra,a new integrable hierarchy is generated by Tu scheme;Expanding models which associated with the existing(2+1)dimensional hierarchy are derived.The nonlinear integrable coupling for the Dirac hierarchy which has Hamiltonian structure are obtained by means of non-semisimple matrix Loop algebra;The bi-integrable coupling of two integrable hierarchies are constructed through semi-direct sums Lie algebra,and the Hamiltonian structure is furnished by the variational identity.In chapter 1,the relevant definition of integrable system is introduced,then a new integrable hierarchy with bi-Hamilton structure was derived from Zero curvature equation;The second chapter presented the method arising the integrable coupling,and proposed the Tu-Andrushkiw-Huang(TAH)scheme which constructed(2+1)dimensional hierarchy.Based an old(2+1)dimensional hierarchy,constructing two kinds of integrable couplings of corresponding(1+1)dimensional integrable hierarchy and obtained the expanding model.In chapter 3,based an generalized Dirac integrable hierarchies with a bi-Hamiltonian structure.the nonlinear integrable couplings hierarchy is obtained by means of semi-simple Loop matrix algebra.Based on the semi-direct sums of matrix Lie algebras,the Hamiltonian structure of the integrable couple system is furnished by the variational identity.Then,extending two kinds of Kaup-Newell(KN)equations in different forms of bi-integrable coupling,and discuss their Hamilton integrability by using operator properties.The last section is devoted to conclusions and discussions.