Exact Solutions Of Some Nonlinear Partial Differential Equations And Their Dynamic Behaviors

Based on the theory of Lie symmetry analysis method,and supplemented by Maple,this thesis comprehensively adopts power series method,MSE method,dynamical system method and B(?)cklund transform method to solve the partial differential equations(systems).To be specific,the main idea of this thesis is: Firstly,based on Lie symmetry analysis method and supplemented by Maple,the vector fields of these equations(systems)are obtained;secondly,the similarity reduction is applied to convert the complex nonlinear partial differentia l equations(systems)to be considered into simpler forms of ordinary differential equations(systems);finally,according to the features of specific ordinary differential equations(systems),suitable methods are selected to study the obtained ordinary d ifferential equations(systems).In this way,it is possible to solve the reduced ordinary differential equation(systems),which is used to obtain the exact solution of the corresponding partial differential equation(systems).The major work of this thesis are as follows:Chapter 1 is the introduction.This chapter mainly introduces the development history and research status of nonlinear partial differential equations(systems),the basic ideas of Lie symmetry analysis method and the main research results obtained in this thesis.This chapter is to collate and review various classic methods,compare,summarize and adopt the advantages of each method,and further use the excellent methods to study more complex nonlinear partial differential equations.In chapter 2,based on the MSE method,some exact solutions of nonlinear partial differential equations(systems),such as STO equations and similar Hirota-Satsuma KdV systems,are constructed.These equations are extensively applied in physics,so their exact solutions are helpful for theoretical and numerical researches.Meanwhile,as for the obtained exact solutions,when different values are taken to parameters,different solitary wave solutions can be obtained correspondingly.Finally,the image of solitar y wave solutions corresponding to equations is presented by the use of Matlab.In chapter 3,by means of(G/ G?)expansion method,the sine-Gordon equations,sinh-Gordon equations and Liouville equations are solved to respectively obta in hyperbolic function solutions and trigonometric function solutions of the equations.This method carries direct and simple advantage in constructing the exact solution of nonlinear partial differential equations.Therefore,it is hoped that this method can be used to construct exact solutions of more complex nonlinear evolution equations.Based on the principle of Lie symmetry analysis,chapter 4 researches the nonlinear transmission line potential equation,so that the vector field of the equation is ob tained,and the results are verified by Maple calculation software.Later,further combined with the principle of homogeneous equilibrium,power series expansion method and(G/ G?)expansion method,multiple groups of exact solutions o f reduced ordinary differential equations are obtained to acquire the exact solution of original equation.In this way,the display solution of the potential equation of nonlinear transmission line is greatly enriched.This is enough to show that,Lie symmetry analysis method play significant role in solving nonlinear partial differential equations(systems).Based on homogeneous balance method,chapter 5 considers and studies a class of fourth-order partial differential equation.Firstly,the B(?)cklund transformation method is used to obtain the exact solutions of this equation.Secondly,Lie symmetry analysis method is adopted and combined with Maple calculation software to obtain the vector field of this equation.Besides,similarity transformation is ut ilized to convert the complicated nonlinear partial differential equations that are usually difficult to solve into the ordinary differential equations can be solved easily.Finally,combined with the expansion method of power series,the reduced ordinary differential equation is solved so as to obtain a series of exact solutions of original equation.In C hapter 6,based on Lie symmetry analysis method,the power series expansion method and the dynamical system method are combined to obtain the exact solution of combined sinh-cosh-Gordon equation.Firstly,the vector field of the equation is obtained by using Lie symmetry analysis method.Secondly,the phase diagram branch of the reduced equation is drawn through Matlab.Finally,the kinetics behavior of the equation is studied via power series expansion method and the dynamic system method,so that the exact travelling wave solution of the equation is concluded.The last part is the summary and the outlook.In terms of main conclusion,all the research results obtained previously were obtained,and then the main innovations of this thesis are concluded;finally,with respect to outlook,some suggestions on how to solve nonlinear partial differential equations(systems),along with the research outlook,are proposed.