Exact Solutions Of Some Nonlinear Partial Differential Equations

The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solution properties in the pure and applied science. Therefore, solving soliton Equations is very important in the theory and in the application. This dissertation, under the guidance of mathematics mechanization and by means of symbolic computation software, considers the exact solutions to the nonlinear partial differential equations. With the help of symbolic computation system Maple, by the application of improved methods of solving equations, some new exact solutions are presented.Chapter1 is to introduce the history and development of the soliton theory, including the origin of the soliton theory, development of the method solving nonlinear equations, also introduced mathematical mechanization and symbolic computation and application.Chapter2 is the application of the F-expansion method to the generalized Hirota -Satsuma coupled equations. First on the F-expansion method steps, and then by the F-expansion method , we get a lot of the new exacted solutions that can not be find in reference .Chapter 3 is to introduce the solving of (2+1)-dimensional Burgers equation by the improved coupling Riccati equations raised by reference [13]. First of all, we describe the steps of the F-expansion method , and then apply the method, many of the new exact solutions of (2+1)-dimensional Burgers equation are obtained.Chapter 4 is to firstly introduce the solving of KdV-MKdV equation by the improved Riccati equation . Secondly, we obtained the like soliton solutions and the periodic solutions of the (2+1)-dimensional generalized shallow-water wave equation which are with variable coefficient, because of the variable coefficient ,the equation will be have more like soliton solutions and periodic solutions.Chapter 5 begins with a brief introduction to solving the nonlinear development equation by an effective method - Darboux transformation method, followed by a new Darboux transformation is presented with the proof, finally, the new Darboux transformation is applied to BK system, then we obtained the new solutions of BK system.