Existence Of Traveling Wave Solutions For Keller-Segel System And KP-MEW-Burgers Equation

The study of traveling wave solutions of nonlinear differential equations is of great significance in physics or biology.The Keller-Segel model is a well-known biomathematical model that describes the phenomenon of biochemotaxis.Using rigorous mathematical methods to study the properties of the Keller-Segel model to understand the laws of organism movement has attracted the attention of many experts and scholars.The Burgers equation is one of the most basic equations in the fields of fluid mechanics,nonlinear acoustics and gas dynamics.As a mathematical model describing the motion phenomena of fluids,the study of Burgers equations is of great value for understanding various physical phenomena and promotes the development of nonlinear differential equations.In this paper,by using the dynamical systems approach,specifically based on geometric singular perturbation theory,combined with Abelian integrals theory,implicit function theorem etc,we study the existence of traveling wave solutions for the Keller-Segel system with nonlinear chemical gradients and small cell diffusion and the KP-MEW-Burgers equation with local delay and small damping.This paper includes three chapters as follows:Chapter 1 briefly introduces the background and significance of this paper,and briefly introduces the main work of this paper and some preliminary knowledge.Chapter 2 considers the one-dimensional Keller-Segel system with nonlinear chemical gradients and small cell diffusion.We first analyze the dynamics of the system by geometric singular perturbation theory.And then we seek an invariant region for the associated traveling wave equation of the system.Finally,we apply PoincaréBendixson theorem to analyse the flow on this invariant region to obtain the existence of traveling pulse solutions for Keller-Segel system.Chapter 3 considers the generalized(2 + 1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers(KP-MEW-Burgers)equation with local delay and small damping.By using geometric singular perturbation theory,the existence of solitary wave and periodic wave of KP-MEW-Burgers equation is obtained.Moreover,the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals and the upper and lower bounds of the limit wave speed are given.In addition,the lower bound and monotonicity of the period of traveling wave solutions when the small positive parameter τ → 0 are also obtained.