The Existence Of Traveling Wave Solutions For The KdV Equations With Time Delay

KdV equation is one kind of the most important nonlinear models. The travelingwave solutions when they exist can enable us to well understand the mechanism of thecomplicated physical phenomena and dynamical processes modeled by these nonlinearevolution equations. The existence of traveling wave solutions of KdV equation is animportant objective.In this paper, by using the geometric singular perturbation theory, the qualitativetheory of ordinary differential equations, the linear chain trick and the dynamical sys-tems, we research the existence of traveling wave solutions for the KdV equations withtime delays. The paper includes four chapters as follows:Chapter1introduces the backgrounds and significance of KdV equations and trav-eling wave solutions, main work of this paper.Chapter2considers the traveling wave solutions of the Burgers-Korteweg-de Vriesequation with time delay. We apply the geometric singular perturbation theory and thequalitative theory of ordinary differential equations to study the homoclinic orbit ter-minated at the origin on a two-dimensional inertial manifold, and obtain the existenceof traveling wave solution. We extend some results in references.Chapter3mainly discusses solitary wave solutions for a generalized KdV-mKdVequation with distributed delays. By employing the geometrical singular perturbationtheory and the linear chain trick to study the homoclinic orbit terminated at the originon a two-dimensional inertial manifold, we establish the existence results of solitarywave solutions when the average delay is sufficiently small, for a special convolutionkernel. Our results generalize and improve some known results.Chapter4is concerned with the traveling wave solution of the fourth order cB-KdV equation. We use the method of dynamical systems, and the geometric singularperturbation theory to study the existence of a heteroclinic connection between twocritical points and establish the existence of traveling wave solutions.