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

The major contents in this paper include: the solutions of the soliton equations and the integrable system. In the second chapter, firstly the generalized Jacobi elliptic function method is further improved by introducing an elliptic function φ(ξξ) as a new independent variableand abundant new traveling wave solutions of the coupled Drinfel' d-Sokolov-Wilson equation are obtained. Secondly, based on the well-known Sine-Poisson equation, a new Sine-Poisson expanding method with constant coefficients or variable coefficients is presented, which can be used to construct more new exact solutions of nonlinear evolution equations. The KdV-mKdV equation and breaking soliton equation are chosen to illustrate this method. In thethird chapter, firstly a new basis of loop algebra A1 is presented and a new isospectral problem is designed from it. It follows that a type of new Liouville integrable hierarchy with five potential functions which posseses bi-Hamilton structure is obtained by using of Tu scheme, which can be reduced to Tu hierarchy. Secondly, a type of new loop algebra A2 is constructed by applying a kind of new commutative operation. Then an isospectral problem is designed from it. It follows that a new integrable hierarchy, which possesses bi-Hamiltonian structure, is obtained by using of Tu scheme. Thirdly, a new loop algebra x with 3 M dimensions is constructed by use of some properties of exterior algebra, which is devoted to establishing a new isospectral problem. As its application, a multi-component integrable system similar to the TC hierarchy is obtained, whose reduction case presents a multi-component KdV equation. At the last, a type of new loop algebra GMis constructed by taking use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy. In the fourth chapter, we obtained a kind of expanding integrable model of a known hierarchy and proposed a direct method for the expanding integrable model.