Lie Symmetry Analysis,Exact Solution And Dynamic Behavior Of Several Kinds Of Partial Differential System

In this paper,the theory of lie groups,planar dynamical system theory are used to study three kinds of partial differential equation respectively.At first,with the theory of Lie group,we studied two classes of Schrodinger equation and find the symmetry and exact solutions;Then by using planar dynamical system theory,we studied a class of partial differential equations of dynamic branch and branch phase portraits;finally used the method of modified CK method and the extend trial function method to discuss the fifth-order KDV equations with variable coefficients,then we get the appropriate similarity transformation and a series of corresponding exact solutions and phase portraits.In the first chapter,two kinds of Schr?dinger equations of CNS and NLS are studied by complex envelope transformation and Lie symmetry analysis,the symmetric and reduced equations of the two kinds of equations are obtained.and then many exact solutions are obtained by the symmetries and reduced equations.Then the new solutions of the power series form are obtained by analyzing the higher-order reduced equations of the two kinds of equations.In Chapter 2,the bifurcation and phase diagrams of a class of partial differential equations are given by using the theory of plane dynamical systems,the bifurcation behavior of the traveling wave solutions of the equations are studied.Furthermore,the parametric expressions of the possible exact solutions are studied by analysis of phase portraits.Finally,the results are discussed.In Chapter 3,the equivalence relation between the fifth-order KDV equation with variable coefficient and the corresponding constant coefficient equation is given by using the modified CK direct method.Then introduced the ETEM briefly and obtain some new types of travelling wave solutions of fifth-order KDV equation with variable coefficient.