Lie Symmetry Analysis And Exact Solutions For Several Kinds Of Nonlinear Evolution Equations

Nonlinear evolution equations(NLEEs)are common partial differential equations that can be used to explain nonlinear phenomena in various branches of physics and engineering science,such as fluid mechanics,nonlinear dynamics,fiber optics and acoustics.It is an indispensable part of theory and applied science to analyze physical phenomena through mathematical models.These models are usually nonlinear partial differential systems.These systems are complementary to the theory of describing complex physical phenomena in essence.Therefore,nonlinear partial differential equations(NLPDEs)have been developed to the cornerstone of many studies.The thesis is mainly based on the Lie symmetry analysis theory,combined with(G'/G2)-expansion method,Painleve analysis,power series method and adjoint equation method,the symmetries,exact solutions and conservation laws to some nonlinear evolution equations are discussed with the mathematical symbol computing system Maple.The main content of the first chapter is the research background and significance of this thesis and the overview of research methods used.This thesis introduces the source and development of partial differential equations,the existing research results and the basic idea and preliminary knowledge of Lie symmetry analysis.At last,the content of the thesis is described.In the second chapter,the Camassa-Holm equation with constant coefficients is solved based on the Lie symmetry analysis method and the(G'/G2)-expansion method.The power series solution,hyperbolic function solution,trigonometric function solution and rational function solution of the equation under different conditions are discussed respectively.Using the symbolic calculation tool Maple,the three-dimensional images of solitary wave solutions are drawn under special parameter values.In the third chapter,two classes of fifth-order nonlinear evolution equations with constant coefficients are considered by using the Lie symmetry analysis.The symmetries and optimal systems under different conditions are obtained by means of the symbolic calculation tool Maple.Then,using the symmetry reduction,the partial differential equations are reduced to ordinary differential equations.Moreover,the exact solutions to reduced equations are solved,then the exact solutions in different forms to original equations are obtained including the power series solution.Finally,according to the adjoint equations and symmetries,conservation laws of two equations are provided,respectively.In the fourth chapter,the generalized time-dependent coefficients Gardner equation is investigated by applying Painleve analysis,Lie symmetry analysis and power series method.Firstly,the integrable condition and exact solutions are obtained by Painleve analysis and vector fields of the equation based on Lie symmetry method.What's more,the power series method is applied to solve the exact solutions to the reduced ordinary differential equations.Some exact solutions of the original equation are obtained,and the explicit solutions to the time-dependent Gardner equation are extended.They have certain significance for complex physical motio.The fifth chapter is the summary and outlook.Firstly,the research contents of the thesis are summarized,and the results are briefly described.Then the future research work on nonlinear evolution equations is prospected.