几类高阶非线性行波方程的精确解,分支和复杂动力学研究

项目名称： 几类高阶非线性行波方程的精确解,分支和复杂动力学研究

项目编号： No.11471289

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈凤娟

作者单位： 浙江师范大学

项目金额： 70万元

中文摘要： 应用动力系统方法研究几类高阶非线性行波方程的精确解和动力学性质。对某些4维可积系统发现系统的精确周期解、拟周期解和同宿、异宿环族。对扰动的可积系统研究同宿、异宿环族的持续性问题和混沌动力学，尝试秩为1的映射理论可能的发展和应用。对于许多具有重要物理意义的非线性波方程，如藕合的非线性Schrodinger方程, 三次和五次Ginzburn-Landan方程, 藕合的广义KdV方程，反应-扩散-对流方程等，研究二维和高维非可积行波系统的解的动力学性质，特别是系统的周期解、拟周期解和同宿、异宿分支和新的混沌动力学。

中文关键词： 微分方程；行波解；非线性系统；定性理论；分支理论

英文摘要： For some high-order nonlinear wave equations, by using the method of dynamical systems, we investigate the dynamical behavior for their corresponding travelling wave systems. Especially, for some four-order integrable systems, we may find interesting exact explicit priodic and quasi-periodic solutions, homoclinic or heteroclinic orbit families, as well as some unbounded travelling wave solutions. For perturbed integrable systems, we consider the persistence problems of homoclinic family and heteroclinic family. In addition, we discuss the chaotic dynamics for these systems. We try to develop the chaos theory of rank-one mapping posed by Prof. Wang, Q. D. etc. to the above four-order travelling wave systems. For a lot of nonlinear wave systems, such as coupled nonlinear Schrodinger equations, cubic and quintic Ginzburn-Landan equations, coupled generalized KdV equations, diffusion-convection-reaction equations, we investigate the dynamical behavior of solutions of their travelling wave systems and find periodic solutions, quasi-periodic solutions, the existence of homoclinic and heteroclinic manifolds.

英文关键词： differential equation;traveling wave;nonlinear system;qualitative theory;bifurcation theory