A Constructive Method For Solving Exact Solutions Of Some Nonlinear Evolution Equations

In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of "AC=BD" theory of Professor Zhang Hongqing, we consider some methods seeking exact solutions to nonlinear partial differential equation(s) arising from the fields of elasticity, fluid mechanics, aerodynamics, plasma physics, biophysics and chemical physics.Chapter 1 of this dissertation is devoted to investigating the theory and application of mathematics mechanization, reviewing the history and development of the soliton theory. In addition, some domestic achievements and foreign ones on the subject are presented.Chapter 2 concerns the construction of exact solutions to nonlinear partial differential equation(s) under the uniform frame work of "AC=BD" theory. The basic theory and application regarding "AC=BD" model and the construction of operators C and D are introduced.Chapter 3 introduces some methods seeking exact solutions for the nonlinear evolution equation, such as inverse scattering method, symmetry reduction method, Backlund transformation, Darboux transformation , Hirota bilinear method, Painleve analysis method, AC = BD model, and so on. We present a new elliptic equation rational expansion method to uniformly construct a series of exact solutions for nonlinear partial differential equations. As an application of the method, we choose the (2+1)-dimensional Burgers equation to illustrate the method and successfully obtain some new and more general solutions.