AC=BD Theory And Its Application To Exactly Solving Burgers Equation

In this thesis, under the guidance of mathematics mechanization and by means of symboliccomputation software Maple, the AC=BD theory put forward by Prof. Zhang Hongqing isintroduced to find exact solutions to Burgers equation. The methods presented for constructingthe exact solutions to Burgers equation can also be applied to other nonlinear partial differentialequations(NPDEs) in the theory of soliton.The structure of this paper is as follows:Chapter 1 is devoted to investigating the history and development of soliton theory, theideas of the mathematics mechanization and computer algebra. In addition, some methodspresented by some domestic achievements and foreign ones for constructing the exact solutionsto NPDEs are introduced, such as inverse scattering method, B(?)cklund transformation andDarboux transformation, bilinear method, AC=BD model, and so on.Chapter 2 considers the construction of exact solutions to NPDEs under the instructionof the AC=BD theory introduced by Prof. Zhang Hongqing. Some examples are chosen toillustrate how to use the principle and the application range. And the methods of constructingthe operators of C and D are introduced.Chapter 3 introduces seeking exact solutions of Burgers equation which is significant tomathematic and physics. To construct more new exact solutions of Burgers equation, we pro-pose extended Riccati equation rational expansion method and Jacobi elliptic function rationalexpansion method. And respectively we take coupling high-dimensional Burgers equation and(2+1)-dimensional Burgers equation for example to illustrate the effectiveness of the methods.