Seeking For Some Solutions Of Partial Differential Equation By Bilinear Method

With the development of modern mathematics, the study of nonlinear science and complex systems has become a major focus in the development of science. A large num-ber of nonlinear systems proposed have been widely applied to geophysics, aerospace, marine science, weather prediction, natural disaster warning, etc.. Althought it is very difficult to solve nonlinear systems thoroughly, even sometimes it is impossible, for some special nonlinear systems mathematicians have developed many effective methods:vari-able separation approach, Hirota direct method (also known as the bilinear method), the inverse scattering method, Darboux transformation, Backlund transformation and so on. Among them, Hirota direct method is an effective one, which can solve not only integrable equations but also non-integrable ones. On the basis of thoroughly studying the inherent properties and various applied techniques of the Hirota bilinear method, we obtain the bilinear form of a type of variable-coefficient Schrodinger equations successfully and some general one-soliton solutions and some two-soliton solution. At the same time, the Hirota bilinear method is extended to obtain the bilinear form and some solutions of a supersymmetric seventh-order KdV equation. Some useful conclusions are drew, which reveal that the bilinear method can be more widely used in nonlinear science.The thesis is arranged as follows:Chapter 1 Introduction:Overview the history and progress of soliton equation briefly and several methods for solving soliton equation are introduced in this chapter.Chapter 2 Hirota bilinear method and its application:Introduce comprehensively and systematicly to the basic facts of the bilinear differential operator and how to bilin-earize nonlinear differential equations, specifically solve the NLS equation with various parameters and give the corresponding bilinear process. Various types of soliton solutions are analyzed and summarized for a kind of bilinear form of the NLS.Chapter 3 Supersymmetric Hirota bilinear method and its application:Extend the application range of the bilinear method to super-equations and introduce the specific definition of the supersymmetric bilinear differential operator, obtain some solutions of the 7-order supersymmetric KdV equation and analyse their practical applications.