Application Of Symmetry Analysis And Dynamical System Methods To Several Kinds Of Nonlinear Equations

In this paper,three types of nonlinear equations are studied by symmetry analysis and dynamical system method.Firstly,a class of nonlinear shallow water wave equations is discussed by using Lie group theory and dynamical system method,and a series of exact solutions are obtained.Secondly,the(2+1)-dimensional dissipative long wave system is studied by symmetry analysis and plane dynamic system method,and the(2+1)-dimensional nonlinear partial differential system is transformed into(1+1)-dimensional(ODE).By studying(1+1)-dimensional(ODE),the exact solutions of(2+1)-dimensional dissipative long wave system are obtained.Finally,use the invariant subspace method to learn the nonlinear water wave equation,the exact solutions and corresponding images are given.In Chapter 1,the properties of a class of nonlinear shallow water wave equation is analyzed by using the knowledge of Lie group.Secondly,using the dynamic system method,the partial differential equation is transformed into ordinary differential equation.Combining with the special properties of ordinary differential equation,the corresponding the bifurcations and phase portraits are given,and the exact traveling wave solutions of the equation are obtained.Finally,the reduced equation is analyzed by power series method,and new exact solutions of the equation are obtained.In Chapter 2,the vector field,Lie brackets and the symmetry groups of(2+1)-dimensional dissipative long wave system are obtained by Lie symmetry analysis.Using traveling-wave transform(2+1)-dimensional nonlinear system is transformed into(1+1)-dimensional nonlinear equation,using qualitative theory of planar dynamical systems and bifurcation theory,the(1+1)-dimensional nonlinear equation into(ODE),and discussed under different parameter spaces of the bifurcations and the corresponding phase portraits,according to the phase portraits the(2+1)-dimensional dispersive long-wave system of solitary wave solutions,periodic solutions and kink wave solutions(anti-kink wave solutions)are obtained.Meanwhile,the adjoint equations and conservation laws of(2+1)-dimensional dissipative long wave system is studied.In Chapter 3,the invariant subspace classification of a class of nonlinear water wave equation are given,then some exact explicit solutions to the nonlinear equation are provided by using the invariant subspace method.This method is essentially a dynamical system method,and its core step is to transform a nonlinear partial differential equation(PDE)into ordinary differential equation(ODE)systems,then by solving the ODE systems,the exact solutions to the nonlinear PDE are obtained.