Lie Group Analysis,Exact Solution And Conservation Laws For Several Kinds Of Nonlinear Partial Differential Equations

In this paper,The Lie group analysis theory was employed to study three kinds of nonlinear partial differential equations: a class of four order partial differential equation with variable coefficients,the generalized(3+1)-dimensional ZK equation and the five order dispersion equation.Firstly,we obtained Lie point symmetries of the above equations.Then we obtained the group invariant solutions and reduced equations.Moreover,we solved the reduced equations.Furthermore,we gained exact solutions of the original equations,including the trigonometric function solutions,rational function solutions and Jacobi elliptic function expansion solutions.Finally,we gave adjoint equation and the conservation laws of the first two categories of partial differential equations according to the Lie point symmetry that obtained.In Chapter 1,using the theory of Lie group analysis,we studied a class of four order partial differential equations with variable coefficients.Classified of variable coefficients according to symmetric constraint condition,we gained all the Lie point symmetries and several representative partial differential equations.Those partial differential equations were reduced to ordinary differential equations by selecting appropriate transformation.Then,we combined with the expansion method and elliptic function expansion method,the exact solutions of original equation obtained after seeking solutions of reduced equations.Further,the conservation laws was given for this class of this partial differential equations.In Chapter 2,we solved the generalized(3+1)dimensional ZK equation based on the Lie group analysis theory.First of all,we got all the Lie point symmetries of generalized ZK equation by prolonging vector field.Then,we selected the appropriate similarity transformation,the(3+1)dimensional ZK equation was directly reduced to(1+1)dimensional partial differential equation and ordinary differential equations.Further,the solutions of the reductive equation gained with the help of several auxiliary equations.Thus,the exact solutions of the original equation were obtained.Meanwhile,the adjoint equation and the conservation laws of the generalized(3+1)dimensional ZK equation were obtained according to the Lie point symmetry.In Chapter 3,five order dispersion equation with variable coefficients was explored with the aid of the modified CK method.We reduced five order dispersion equation with variable coefficients to equation with constant coefficients.Then,combined with Lie group analysis method,we solved five order dispersion equations with constant coefficients.Further,we obtained the exact solutions of the original equation,including precise power series expansion solutions and exponential function expansion solutions.In conclusion,the main thought of this paper was to reduce the complex partial differential equations to ordinary differential equations or less variable equations by using Lie group analysis theory.With the help of auxiliary equations,we could get exact solutions of the original equation.