Backlund Transformation In Solving Nonlinear Partical Differential Equation

With the development of nonlinear science , solving nonlinear partial differential equations become an important part of nonlinear problems. Many mathematicians and physicists have done a lot of work. For describing the laws and property of the complex movement more accurate and objective . And now to research the variable coefficient equation is especially important , but it is so difficult to solve the variable coefficient equations . There is not a unified method to solve them. This paper obtains a series of exact solutions by doing new unknown function transformation of the need solutions of the equation and Backlund transformation.Chapter 1 is to introduce the history and development of the soliton theory ,mathematics mechanization and applications of symbolic computation, the development of the exact solutions of nonlinear evolution equations. Introduce several methods—Darboux transformation,bilinear solution,homogeneous balance principle. The main results of this dissertation are introduced at the end.Chapter 2 introduce the history and development of the Backlund transformation firstly. Based on the equivalence of three forms, transform Backlund transformation of Darboux form to Backlund transformation of bilinear form through a spectral problem. Then based on the homogeneous balance principle ,bring up a new auto- Backlund transformation to solve equations and popularize itChapter 3 use the method bring up from Chapter 2 to solve a the(2+1) dimensional Variable Coefficient Equation, we solve it by doing new unknown functions transformation of the need solutions of the equation and performing mathematical calculations to obtain a series of exact solutions, which contain solution like solutions and rational solutions. Some exact solutions include arbitrary functions, when these arbitrary functions are taken as some special functions, these solutions posses abundant structures. Then the method is used in solving variable coefficient KdV equation,Boussinesq equationΠand KdV-mKdV equation.