Symmetry And Dynamic Properties Of Several Types Of Partial Differential Equations

In this paper,utilizing symmetry theory and the bifurcation theory of dynamical systems,several problems related to nonlinear models in mathematical physics equa-tions have been studied in this dissertation.The study mainly includes the following four aspects:Lie symmetry theory;optimal system;fractional differential equations and dynamics theory.The specific chapters are arranged as follows:In chapter 1,an introduction is devoted to the research background and the current situation related to this dissertation,which includes symmetry theory,optimization the-ory,fractional differential equation theory and bifurcation theory of dynamic systems.The main work of this dissertation is also illustrated.In chapter 2,based on the analysis of the symmetry theory,and by utilizing the classical Lie group method the Lie symmetry,Lie algebra and group invariant solutions of the(2+1)-dimensional Bogoyavlenskii equation are studied.By reducing the equation with the above symmetries,some new exact solutions to the equation are obtained.Fi-nally,Ibragimov integrated the idea of the adjoint equation and Noether's theorem,and used the adjoint equation method to construct the conservation law of the Bogoyavlenskii equation.Through calculation,we can find that this method can be used to calculate new conservation laws for any differential equations.In chapter 3,based on the symmetrical theory studied in Chapter 2,subgroups in the concomitant sense are classified and an optimized system is proposed;and existing methods for constructing optimal systems are also presented here in detail,including Ovisiannikov's theory,Olver's theory and direct construction theory.The three theories are demonstrated one by one through the KdV-like equation.By contrast,we can find the superiority of using the direct construction theory,and we then use this method to study the one-dimensional optimization system and similarity reduction of the Harry Dym equation.In chapter 4,the complete algebra of Lie point symmetries for the class of time fractional weakly coupled Kaup-Kupershmidt equation are derived.With the classical Lie symmetry method,the associated vector fields are obtained which in turn can be used to reduce the equation.In chapter 5,combining the qualitative theory of differential equations with the bifur-cation theory of plane dynamical systems,using dynamic system method,the combined form of KdV and KdV-like equations of ??1 and all bifurcations and the phase por-traits under different parameter areas are studied.The smooth solitary wave solution,kink(inverse kink)wave solution and smooth periodic wave solution and non-smooth traveling wave solutions(such as peakon,cuspon,and periodic cusp waves)are obtained respectively.Finally,exact solutions are studied,with a numerical simulation presented.In chapter 6,the summary and discussion of this dissertation are presented and the outlook of future work is provided.