Researches On The Integrable Systems And Exact Solutions Of The Soliton Equations

Continuous and discrete integrable systems and their integrable expanding models and the exact solutions of the nonlinear equations are presented in this paper.In the first chapter, historical origin and some researches of soliton theory together with its research meaning are presented.In the second chapter,first,integrable couplings of the KdV hierarchy is obtained by using of the new subalgebra of the loop algebra,then the Hamilton structure of the above system is given by the quadratic form identity.Second,a(2+1)-dimensional Li hierarchy is generated from one of reduced equations of self-dual Yang-Mills equations.With the help of a proper loop algebra,the Hamilton structure of its expanding integrable couplings is worked out by using the quadratic form identity,which is Liouville integrable.Third,a corresponding multi-component Li hierarchy is given.In the third chapter,a new way is presented for discrete integrable expanding models with the help of two semi-direct sum Lie algebras.As its applications,two discrete integrable expanding models associated with the lattice equation are worked out.The approach can be used to study other discrete integrable expanding models of the discrete hierarchies of solition equations.In the fourth chapter,by using the homogeneous balance method,we obtain the exact solutions of the(2+1)-dimensional Burgers equation and its corresponding pictures.