Several Kinds Of Equations Integrable Coupling System And Darboux Transformation Solution

In this article, we discuss the following three aspects: the first problem is to construct the new spectral matrix of the integrable coupling systems by using the extended lax pair method; the second prolem is to obtain the integrable coupling Hamilton structure by the variational trace identities; the third problem is to solve the integrable coupled soliton equation by Darboux transformation method.In the second chapter, we study on the continuous WKI soliton equation and the variable coefficients AKNS soliton equation. By the method of extending Lax pairs, we obtain the integrable coupling WKI soliton equation and AKNS soliton equation with variable coefficients. We firstly consider the WKI soliton equation by constructing a special spectral matrix  and  to generate the nonlinear part and the new extension WKI integrable coupling soliton equation. Then we consider the constant coefficients AKNS soliton equation and variable coefficients AKNS soliton equation, and achieve their integrable coupling systems by the method of extending the Lax pairs. The integrable coupling AKNS soliton equations with variable coefficients are signality. Finally, we obtain the Hamilton structures of the WKI soliton equation and the variable coefficients AKNS soliton equation.In the third chapter, we construct the integrable coupling system of Toda lattice soliton equation and the new discrete soliton equation, then we obtain the corresponding integrable coupling equations by the method of Darboux transformation. We firstly consider the discrete Toda lattice soliton equation. By using the Lax pairs U,V ∈sl(8), and gain the new discrete integrable coupling system. Through comparing, we obtained the Hamilton structure of the soliton equation integrable coupling by the variational trace identities. Finally, we solve some discrete integrable coupling equations by using the methed of Darboux Transformation. For two new discrete Lax pairs and corresponding discrete soliton integrable coupling equations, we construct two kinds of different Darboux transformation method to acquire the solutions of the discrete integrable coupling equations.