Extended Symmetry Transformation And Exact Solutions For Several Partial Diferential Equations

For a long time, it is a difcult but very important research subject for solving partialdiferential equation. Lots of scholars in this area have done a lot of excellent work and developeda variety of efective methods to solve diferential equation. However, because of the complexityof the nonlinear diferential equations, these methods didn’t completely solve the problem. It iswidely applied for extended symmetry transformation method and the generalized sub-equationexpansion method to construct exact analytical solutions. By the two methods, we can getmore rich solutions which have new physical significance. Based on the two methods mentionedabove, we have discussed the exact solutions of the nonlinear equation in this paper.The thesis is arranged as follows:Chapter1. Several classic methods for obtaining exact solutions of nonlinear partial difer-ential equations are introduced in this chapter.Chapter2. Firstly,we introduce extended symmetry transformation method and actwith (3+1)-dimensional generalized Schro¨dinger equation. Then,based on the idea of theextended symmetry, we can get an extending symmetry group and related finite transformation.By the solutions of the standard constant coefcient (3+1)-dimensional Schro¨dinger equation,we can construct the various solutions of the diferent (3+1)-dimensional generalized Schr¨odingerequation.Chapter3. Firstly, based on the generalized sub-equation expansion method, we obtian thesolutions of the (2+1)-dimensional generalized variable-coefcients Gross-Pitaevskii equation,and discuss the nonlinear dynamics of the chirped soliton pulses under the diferent regimes ofsoliton management. In addition, we also investigate the solutions of the generalized variable-coefcient cubic-quintic nonlinear Schr¨odinger equation with the help of the generalized sub-equation expansion method.