Symmetry Reductions And Exact Solutions Of A Number Of Nonlinear Partial Differential Equations

Based on the important role of symmetry in the investigation of partial differential equations,and the urgent need of solving nonlinear partial differential equations,this dissertation studies some nonlinear partial differential equations with physical meanings and realistic backgrounds.Using the Lie group method and computer symbolic computation,this paper investigates the symmetry?symmetry classifications?symmetry reductions and exact solutions.By means of traveling wave transformation,we discuss some new exact traveling wave solutions of the variable coefficient nonisospectral KdV equation and the complex coupled KdV equations.Partial differential equations are turned into some simple ordinary differential equations by this way.For the generalized transformation of Klein-Gordon-Zakharov(gtKGZ)equations?the(2+1)-dimensional variable coefficient Boiti-Leon-Pempinelli equations and the(2+1)-dimensional variable coefficient BroerKaup equations,this paper researches other transformation forms of the independent variables by the classical and nonclassical Lie methods.This paper consists of six chapters:Chapter 1 mainly introduces domestic and foreign research status about solving nonlinear partial differential equations,and provides some background knowledge and preliminaries.Chapter 2,based on the idea of the homogeneous balance method,transforms the variable coefficient nonisospectral KdV equation and the complex coupled KdV equations into the ordinary differential equations by the auxiliary functions method.Furthermore,we apply the Maple software to get some exact solutions of these equations,including Solitary-like solutions,trigonometric function solutions,Jacobian elliptic function solutions and rational solutions.Chapter 3,using the classical Lie method,investigates the symmetry classifications and symmetry reductions of the gtKGZ equations.The key technology is collating and solving of linear overdetermined equations,whose processes can be solved by Maple.Chapter 4 discusses seven symmetries of the(2+1)-dimensional variable coefficient Boiti-Leon-Pempinelli equations by the classical Lie method.Chapter 5 derives symmetry reductions of the(2+1)-dimensional variable coefficient Broer-Kaup equations by means of the nonclassical Lie method.The last chapter concerns the summary for the whole work of this paper,as well as the prospect for the next work.