Lie Group Theory And Exact Solutions Of Three Types Of Nonlinear Evolution Equations

In this thesis,the Lie group method is used as the main research tool.The constructive auxiliary function expansion method is used as an auxiliary method,together with the mathematical software Maple,to investigate three nonlinear development equations that are widely used in various fields such as wave propagation problems,solid mechanics,and fluid kinematics.It includes a class of generalized dispersion equations,a modified fifth-order nonlinear evolution equation,and a class of generalized Burgers-Korteweg de Varies(abbreviated as Burgers-Kd V)equations.Various new exact solutions of these equations are eventually derived.In the first chapter,a class of generalized dispersion equations is investigated by applying the Lie group.In this thesis,we first find the generating elements of these equations and construct an optimal system for the one-dimensional Lie algebra.Then,we uses the resulting symmetry to approximate the original equation into various types of ordinary differential equations.Finally,it sets the solutions in two different forms and obtains some different kinds of exact solutions of this dispersion equation including traveling wave solutions and trigonometric solutions.It also gives the corresponding images of the solutions with the help of Maple to describe their properties.In the second chapter,a generalized fifth-order nonlinear evolution equation is studied by using the Lie group method.First,the vector field of the equation is obtained by using the Lie group method.Then,the original equation is approximated by the vector field and the optimal system.Finally,a series of explicit exact solutions are derived and images of various types of solutions are given using two auxiliary function expansions and the chi-square method.In the third chapter,a generalized class of Burgers-Kd V equations is studied using Lie group theory.Firstly,we find the Lie point symmetry of this equation and construct an optimal system for the one-dimensional Lie algebra.Then this paper uses the Lie point symmetry to obtain the group invariant solution of the original equation.Finally,it gives some new traveling wave solutions and trigonometric solutions of this equation in conjunction with the group invariant solutions and describes the properties of the solutions.