Studies On Bifurcations And Exact Solutions Of Nonlinear Shallow Water Wave Equations

In this dissertation,By using the method of dynamical systems,we research the bifurcations and exact solutions of some nonlinear shallow water wave equations derived from physical problems.The rich dynamic properties of these nonlinear models are revealed.This dissertation is divided into seven chapters.In chapter 1,we summary the historical background,research developments,main methods and achievements of nonlinear shallow water wave equations.Then,we introduce some preliminary knowledge of dynamical systems and the basic mathematic theory and main results of three-step method.In chapter 2,a shallow water wave model is used to introduce the concepts of peakon,periodic peakon and compacton.Traveling wave solutions of the shallow water equation are presented.The corresponding traveling wave system is a singular planar dynamical system with one singular straight line.By using the method of dynamical systems,bifurcation diagrams and explicit exact parametric representations of the solutions are given under different parameter conditions.Periodic peakon solution and quasi-peakon solution are not weak solutions in the sense of distributions,but smooth classical solution with two "time scales".At a peak point,the profile of the periodic peakon solution or quasi-peakon solution looks like nonsmooth,but actually the peak point set is merely a fake-scale profile:it is indeed locally smooth.In chapter 3,we consider the traveling wave solutions for a shallow water equation modeling surface waves of moderate amplitude.The corresponding traveling wave systems are a singular planar dynamical systems with one singular straight line.On the basis of the theory of the singular traveling wave systems,we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions,as well as periodic peakon solutions.We show the existence of compacton solutions of the equation under different parameter conditions.In chapter 4,we consider the Burgers-?? equation.It can be regarded as a model for nonlinear shallow water wave dynamics.By using the method of dynamical systems,we obtain bifurcations of the phase portraits of the traveling wave system under different pa-rameter conditions.Corresponding to some special level curves,we derive possible exact explicit parametric representations of solutions(contain periodic wave solutions,peakon solutions,periodic peakon solutions,solitary wave solutions and compacton solutions)under different parameter conditions.In chapter 5,we consider the Biswas-Milovic equation with F(|q|2)= ?|q|4 + ?|q|2.By using the method of dynamical systems,we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions.Corresponding to some special level curves,we derive possible exact explicit parametric representations of solutions.These solutions include solitary wave solutions,kink and anti-kink wave solutions,periodic wave solutions and compactons.In chapter 6,for a class of nonlinear diffusion-convection-reaction equations,the corresponding travelling wave systems are well known Lienard systems.We first intro-duce Chiellini's integrability condition and the first integral of Lienard system under this condition.Then,by using the method of dynamical systems,we discuss dynamical behavior and some exact solutions of travelling wave systems for generalized damped sine-Gordon equation and the Burgers equation with one-side potential interaction.For the last equation,corresponding to two families of heteroclinic orbits connecting two nodes of the travelling wave system,the existence of uncountably infinite many global monotonic and non-monotonic kink wave solutions is discussed.Under some parametric conditions,exact explicit parametric representations of the monotonic and non-monotonic kink wave solutions are given.In chapter 7,the summary of this dissertation and the prospect of future study are given.