Studies On Exact Traveling Wave Solutions And Existence Of Solutions For Some Nonlinear Wave Equations

This thesis consists of two parts.In the first part,the exact traveling wave solutions of two kinds of nonlinear wave equations are studied by using planar dynamical system theory and singular traveling wave method.These equations are complex GinzburgLandau equation and Raman soliton model respectively.Applying the traveling wave transformation,these equations are transformed into planar dynamical systems,and the bifurcation of phase portraits is drawn by using the bifurcation theory,and various exact explicit solutions of the equation are obtained.In the second part,the bifurcation theory,geometric singular perturbation theory and Melnikov method are used to study the existence of traveling wave solutions of perturbed Degasperis-Procesi(DP)equation and KdV-Burgers-Fisher(KBF)equation with distributed delay.The persistence of the solitary wave solutions of these equations are proved under small perturbations.An outline of the major works is as follows.In Chapter 1,we introduce the research background of the thesis and the related nonlinear wave equation,focusing on the water wave equation,the reaction-diffusion equation and the significant soliton equations in mathematical physics.Furthermore,we also introduce the research background and present situation of nonlinear wave equation with perturbation.In Chapter 2,some preparatory knowledge is introduced,including singular traveling wave method,geometric singular perturbation theory,Melnikov method,etc..In Chapter 3,using the bifurcation theory of the planar dynamical system,we study the exact solutions of the complex Ginzburg-Landau equation which is a popular model in optical fiber communicationin.All possible exact explicit parametric representations of traveling wave solutions are given under different parameter conditions,including the solitary wave solutions,periodic wave solutions,compacton solutions,pseudo-peakon solutions and periodic peakon solutions.In Chapter 4,we investigate the exact solutions of the Raman soliton model with anti-cubic nonlinear in optical metamaterials.By travelling wave transformation,the model is transformed into a singular planar dynamical system with three singular straight lines.Using the bifurcation theory of dynamical systems,under different parameter conditions,bifurcations of phase portraits are studied.More than 30 exact explicit solutions are derived,such as periodic wave solutions,solitary wave solutions,kink and anti-kink wave solutions,periodic peakons and peakons solutions as well as compacton solutions.Compared with the previous work,some new forms of solutions are found under more general parameter conditions,such as periodic peakons,peakons,compacton solutions,etc..In Chapter 5,we study the DP equation,which is a model of shallow water waves.Firstly we establish the existence of solitary wave solutions for the DP equation without perturbation.Then we analyze the persistence of solitary wave solutions of perturbed DP equation by using the geometric singular perturbation theory and Melnikov method.It is proved that the perturbed DP equation also has a homoclinic orbit,by the relation between solitary wave and homoclinic orbit,that is,the perturbed DP equation possesses a solitary wave solution.In Chapter 6,we study the traveling wave solution of KBF equation with distributed delay.The existence of solitary wave solutions of KBF equation is proved by using geometric singular perturbation theory and Melnikov method.Then,we give the expression of Melnikov integral and calculate the zero point exactly.In Chapter 7,we summarizes the work of this thesis and looks forward to some future ideas.