Integrable Expansion Models For Two Types Of Soliton Equation And Wronskian Solutions Of A Class Of Nonlinear Equations

The major contents in this paper include:two kinds of expanding integrable system of soliton equation and searching the solutions of soliton equations by using of Hirota's bilinear method.In the first chapter, historical origin and researchs summarization of soliton theory together with its research meaning are presented. In the second chapter, a new loop algebra A2 is constructed according to a new bracket operation and new isospectral problem is devised according to the given loop algebra A2. As its applications, a new integrable hierarchy is obtained which can be reduced to NLS hierarchy and MKdV hierarchy. The integrable couplings of the integrable hierarchy are obtained by using of the expanding loop algebra G of the loop algebra A2. Then, we first construct asset of multi-component matrix Lie algebra and a type of new loop algebra AM-1. It follows that an isopectral problem is presented. By using of Tu scheme, integrable multi-component S-MKdV hierarchy is presented. Furthermore, we expanded the multi-component matrix loop algebra into a large one and work out a type of integrable couplings of S-MKdV hierarchy. In the third chapter, by introducing logarithm transformation and rational transformation, the (2+1) dimensional KdV equation is transformed into bilinear forms and N-soliton solution of the equation is derived by a perturbation method. Then, the Wronskian technique is introduced to the (2+1) dimensional KdV equation and the Wronskian solutions of the soliton equation are generated.