Traveling Wave Solutions For The Perturbed Defocusing MKdV Equation And Generalized BBM Equation

Based on the relationship between the traveling wave solutions of the nonlinear evolution equation and the homoclinic orbits,heteroclinic orbits and periodic orbits of the ordinary differential equations,we study the existence of the traveling wave solutions of a perturbed defocusing mKdV equation and a perturbed generalized BBM equation.By the traveling wave transform,the perturbed defocusing mKdV equation can be transformed into a fourth-order elliptical Hamiltonian function and the topological phase portrait of the system belongs to the case of two saddle cycle.Furthermore,by the singular perturbation theory and Abelian integral analysis method,we study the existence of the kink wave and periodic wave of a perturbed defocusing mKdV equation.For the perturbed generalized BBM equation,the topological phase portrait of the corresponding traveling wave system belongs to the degenerate case of cuspidal loop.Furthermore,by the singular perturbation theory and Abelian integral analysis method,we study the existence of the kink wave and periodic wave of the perturbed generalized BBM equation.In particularly,we prove the the monotonicity of the wave speed,which only depends on the energy level of k.In this paper,there are five chapters in total and the structure of the paper is given as follows:In Chapter 1,we mainly introduce the background,the research status and the main work of the nonlinear partial differential equation in this introduction.In Chapter 2,we mainly introduce the preliminary knowledge in this paper.In Chapter 3,we study the existence of kink wave,anti-kink wave and periodic wave for a perturbed defocusing mKdV equation by using geometric singular perturbation theory.In addition,by analyzing the perturbation of the Hamiltonian vector field with an elliptic Hamiltonian of degree four,the phase portrait of a two saddle cycle is exhibited.Furthermore,we use Picard-Fuchs system to analyze the ratio of Abelian integral and prove that the wave speed is decreasing.The upper and lower bounds of the limit wave speed are given.Moreover,the relation between the wave speed and the wavelength of traveling waves is obtained.In Chapter 4,we study the existence of solitary waves and periodic waves for a perturbed generalized BBM equation by using geometric singular perturbation theory.Furthermore,In addition,by analyzing the perturbation of the Hamiltonian vector field with an elliptic Hamiltonian of degree four,the phase portrait of a cuspidal loop is exhibited.Furthermore,we use Picard-Fuchs system to analyze the ratio of Abelian integral and prove that the wave speed is decreasing.The upper and lower bounds of the limit wave speed are given.Moreover,the relation between the wave speed and the wavelength of traveling waves is obtained.