Qualitative Geometric Analysis Of Traveling Wave Solutions For Two Nonlinear Wave Equations

Nonlinear wave equation is a very important mathematical model,which can be used to describe nonlinear phenomena in nature,which is first introduced to describe the phenomenon of a single bulge wave by Russell.Now this kind of equation attracted extensive attention of mathematical physicists.Nonlinear wave equations,for example Kd V equation,MEW equation,Burgers equation,KP equation,are widely applied to many physical branches,such as hydrodynamics,plasma physics,nonlinear optics and communication.MEW-Burgers equation and KP-MEW-Burgers equation,which are evolved from two or more equations,have been widespread concerned recently.In this dissertation,the dynamical behaviors of traveling wave solutions of one-dimensional MEW-Burgers equation and two-dimensional generalized KP-MEW-Burgers equation are discussed by the qualitative theory of differential equations and KCC theory.The chaotic complexity of the systems under periodic disturbances is analyzed by numerical simulation.The main contents are as follows:In Chapter 1,the research background,significance and existing results of the related research are introduced briefly.MEW-Burgers equation and generalized KP-MEW-Burgers equation are given.The relevant theories of planar differential system are briefly introduced,including Poincar?e compactification technique,some basic concepts and conclusions of KCC theory.In Chapter 2,the dynamical behavior of traveling wave solutions of MEW-Burgers equation is studied.Firstly,MEW-Burgers equation is transformed into an equivalent planar differential system by traveling wave transformation.The stability of the finite singular points and the singular points at infinity of the planar system is studied,the global structure diagrams of the system with different parameters are obtained.By using the relationship between the trajectories near the equilibrium points of the equivalent planar system and the traveling wave of the wave equation,the obtaining results show that within some certain ranges of parameters,there are solitary waves,kink waves(anti-kink waves)and periodic waves in MEW-Burgers wave equation.Secondly,Jacobi stability at any point of trajectory of the system is discussed based on KCC theory,and Lyapunov stability and Jacobi stability of the equilibrium points are compared and analyzed.The results show that for the wave equation propagating to the left,the trajectories(including the equilibrium points)of the corresponding planar system are all Jacobi unstable,and Lyapunov stability and Jacobi stability of the equilibrium points of the system is not completely consistent.In addition,the focusing tendency of the trajectories near the equilibrium points of the system is analyzed by the dynamical behavior of deviation vector.Finally,the numerical results show that the system presents periodic,quasi-periodic and chaotic phenomena under periodic disturbances.In Chapter 3,the dynamical behavior of traveling wave solutions of the generalized KPMEW-Burgers equation is discussed.Firstly,Lyapunov stability of the finite and infinite singular points of the planar dynamical system equivalent to the generalized KP-MEW-Burgers equation is studied,and the global structures of the system under different parameter conditions are obtained.It is found that the generalized KP-MEW-Burgers equation has solitary waves,kink waves(anti-kink waves)and periodic waves under some certain conditions of parameters.Secondly,based on the KCC theory,Jacobi stability of any point of the system trajectory is discussed,and Lyapunov stability and Jacobi stability of its equilibrium points are analyzed comparatively.The obtaining results show that the equilibrium points of the planar system are all Jacobi unstable within a certain parameter range,and at least one of them is Jacobi stable outside the same range of parameters.According to the dynamical behavior of deviation vector,the focusing tendency of trajectories near the equilibrium points is also analyzed.Finally,the numerical results show that the system presents periodic,quasi-periodic and chaotic phenomena under periodic disturbances.In Chapter 4,the summary of this study is given,some further research ideas are put forward.