Billinear Method And Auxiliary Equation Method For Nonlinear Partial Differential Equation

Soliton theory is widely used in many fields, it is closely related to physics, biology and other disciplines. Since many kinds of nonlinear evolution equations can be derived from various fields, obtaining exact solutions of nonlinear partial differential equations has a practical significance and application value. Bilinear method and auxiliary equation method are two direct and effective methods of obtaining the exact solutions of nonlinear evolution equations. Although the two methods have achieved booming developments in recent years, but them need to be further studied.The main starting point is to generalize the bilinear method and auxiliary equation method to some nonlinear evolution equations, and then obtain new exat solutions. In the part of introduction, we introduce the overview of soliton theory and the histories and development prospects of the bilinear method and auxiliary equation method. In order to outline the specific process of solving nonlinear partial differential equations by auxiliary equation method, we take F-expansion method as an example. In the second chapter, we take a variable-coefficient KdV equation as an example to describe the concrete solving steps of bilinear method. Then the bilinear method was generalized and applied to the(4+1) Fokas equation and a new generalized coupling equations. As a result, we obtain new single-soliton solution, double-soliton solution as well as the general representation n-soliton solution, and then simulate the linear transmission of single-soliton solution and the evolutionay plots of the elastic collisions happened between the double-soliton solution and those among the three soliton-solution. In chapter 3 and chapter 4, we presented two new applications of the auxiliary equation method and obtained new solutoins of an equation containing seven order partial derivative term and a variable-coefficient equation.