几类非线性扰动微分方程的行波解

Geometric singular perturbation theory is an important tool and method for studying the traveling wave solutions of differential equations. It uses invariant manifolds in phase space to understand the global structure of the phase space or to construct orbits with desired properties, which is significant for the research of traveling wave solution. This project mainly uses geometric singular perturbation theory, invariant manifold theory, Melnikov function method and asymptotic analysis method to study the following problems:. Firstly, we study some nonlinear dispersive-dissipative equations, reaction-diffusion equations and epidemic disease models. By seeking appropriate invariant manifold and constructing heteroclinic orbits or homoclinic orbits, we obtain the existence of traveling wave solutions or solitary wave solutions. Secondly, by establishing the connection of the traveling wave solution between differential equations with time delay and the corresponding differential equations without delay, we prove the existence and non existence, asymptotic behavior of traveling wave solutions, and find the minimum velocity. Lastly, from the viewpoint of dynamical systems, we use geometric singular perturbation theory to study the solitary wave solutions of perturbed Camassa-Holm equation, give a detailed qualitative analysis and intuitively understand the geometric properties of solitary wave solution. . This project will play an important role in the development of the study of the traveling wave solutions for nonlinear differential equations, geometric singular perturbation theory and applications. It is of great significance.

几何奇异摄动理论是研究微分方程行波解的一种重要工具和方法，通过对相空间上对不变流形研究，理解相空间的全局结构或者构造预期的轨道，对于行波解的研究具有重要意义。本项目主要运用几何奇异摄动理论、不变流形理论、Melnikov函数方法以及渐近分析理论等研究以下问题：. 1、研究几类非线性高阶色散耗散方程、反应扩散方程和传染病模型，寻找合适的扰动不变流形，构造异宿轨或者同宿轨，得到方程行波解或者孤波解的存在性；2、建立带有时滞的方程和对应的不带时滞的方程行波解之间的联系，证明行波解的存在性和不存在性，研究行波解的渐近性，找到最小波速；3、从动力系统的观点，运用几何奇异摄动理论研究具有扰动的Camassa-Holm方程孤波解的性态，并进行详细的定性分析，直观了解孤波解的几何性质。. 本项目将对非线性微分方程行波解和几何奇异摄动理论及其应用的研究发展起促进作用，具有重要的意义。

本项目主要研究带有扰动的浅水波方程和反应扩散方程行波解的动力学性质。几何奇异摄动理论是研究微分方程行波解的一种重要方法，运用这种方法研究行波解的动力学性质具有重要的实践意义和理论意义。本项目研究带有Kuramoto-Sivashinsky扰动的Camassa-Holm方程和Camassa-Holm-KP方程、广义的KP-MEW-Burgers方程、具有Kuramoto-Sivashinsky扰动的Kawahara方程、具有扩散项的广义Nizhnik-Novikov-Veselov方程、高阶KdV方程和扰动浅水波方程; 研究带有小细胞扩散和增长项的Keller-Segel 生物趋化系统，具有饱和效应的Sel’kov模型，非线性Belousov-Zhabotinskii振荡系统, 非局部Lotka-Volterra竞争系统，具有Crowley-Martin功能函数的捕食模型和种群动力系统等。得到行波解的存在性、不存在性、渐近性、图灵班图稳定性等动力学性质。.项目组在Journal of Nonlinear Science, Journal of Differential Equations，Discrete and Continuous Dynamical Systems Series B,Communications on Pure and Applied Analysis, Qualitative Theory of Dynamical Systems, Applied Mathematical Letters以及中国科学数学等期刊上发表论文22篇，1篇论文被Web of Science列为ESI高被引论文。项目负责人获得山东省科学技术奖（自然科学奖）二等奖1项（排名第二），并入选江苏省“333高层次人才培养工程”中青年科技领军人才，共培养15名研究生。本项目对非线性微分方程行波解和几何奇异摄动理论及其应用的研究发展有推动作用，具有重要意义。