几类不可积系统行波解的分岔

项目名称： 几类不可积系统行波解的分岔

项目编号： No.11301043

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 周钰谦

作者单位： 成都信息工程学院

项目金额： 22万元

中文摘要： 近年来，在偏微分方程行波解分岔的研究上出现了很多工作，但大部分主要涉及可积系统。比起可积系统，不可积系统的行波具有更丰富的类型和更复杂的动力行为，除典型的孤波、扭波和周期波外，它还存在多种振荡行波，系统中周期波的产生也常常会伴随孤波或扭波的破裂。但不可积系统不具有首积分，研究起来难度大，通常需要讨论不变流形的存在性、横截性和处理复杂的阿贝尔积分。不仅如此，其孤波和周期波产生的条件也更为苛刻。因而，要搞清楚不可积系统的孤波在什么条件下产生？伴随它的破裂会不会产生周期波？系统中的振荡行波有哪些类型？振荡行为如何？会不会逐渐趋于一种周期振荡？这些都是值得探讨的问题。本项目将尝试克服不可积性带来的困难，利用动力系统的分岔理论、几何奇异摄动理论、Fredholm算子理论和微分几何的一些方法，从动力学的角度去探索和解释不可积系统中各类行波产生、演化和消失的动力行为，并形成一套对其研究行之有效的方法。

中文关键词： 不可积系统；行波；动力系统；分岔；不变流形

英文摘要： In recent decade, many efforts have been devoted to bifurcation of traveling waves of PDEs. But most of them are mainly concerned with the integrable system. As far as the non-integrable system is concerned, it has more types of taveling waves which possess more complicated dynamical behaviour than the ones of the integrable system. For instance, in the non-integrable system, besides the classical solitary waves, kink waves and periodic waves, various oscillatory traveling waves could occur. In addition, and a periodic wave usually arise from breaking of a solitary wave or a kink wave. Because the non-integrable system has not the first integral, we need overcoming more difficulty and applying more techniques and tools to study it, such as discussing the existence and transversality of the invariant manifold and dealing with the complicated abelian integral. Not only that, the existence condition of a solitary wave or a kink wave could be more rigorous. Naturally, many significant questions need us to discuss. For instance, how does a solitary wave arise? Can a periodic wave appear as a solitary wave breaks? What types of oscillatory traveling waves will occur in the system? What oscillatory behaviour do they have? Can they approach a periodic oscillation? In this project, by using the dynamical system meth

英文关键词： Non-integrable system；Traveling wave；Dynamical system；Bifurcation；Invariant manifold