几类不可积系统行波解的分岔

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 methods, geometry singular perturbation theorem, Fredholm operator theory and some techniques in differential geometry, we plan to investigate and explain how various types of traveling waves in non-integrable system appear, evolve and fade away, from a ponit of view of dynamics. Finally, through the series of study, we expect to form a set of effective methods to deal with traveling waves of non-integrable system.

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。

本项目主要针对不可积系统中行波的存在性及其复杂的动力学行为展开探索性研究。项目主要选择了三个有很强物理背景的偏微分方程模型作为研究对象。它们是高度非线性化的(2+1)维Zoomeron方程、两类带有典型Burgers耗散项的不可积模型：KdV-Burgers-kuramoto方程和Benney-Kawahara-Lin方程。研究内容涉及方程的对称约化与显示解；不可积系统低维不变流形的存在性；高维系统在低维不变流形上的约化和逼近；不稳定流形的跟踪；不变流形的横截性与异宿轨的存在性；行波系统的非局部分岔；有界行波在小扰动下的保持性和数值仿真。在国家自然科学基金青年项目的资助下，我们的研究取得了如下成果：1、较完整地获得了(2+1)维Zoomeron方程的显示有界和无界行波解。2、获得了两类不可积系统的对称约化和级数解。3、证明了两类不可积系统的行波系统中二维不变流形的存在性，并获得了原高维系统在此不变流形上的约化和逼近。4、讨论了二维不变流形的横截性和两类异宿轨的存在性。5、证明在小扰动下异宿轨的保持性，从而获得了原系统两类扭波的存在性。6、讨论了系统的同宿分岔和Poincare分岔，证明了原系统中孤波和周期波的存在性，并利用数值仿真验证了相关结果的正确性。本项目的探索性研究表明最初设想的研究路线是行之有效的。我们的结论是：采用动力系统的分岔理论、几何奇异摄动理论和Fredholm算子理论相结合的研究路线，从行波系统相空间几何结构的角度去研究不可积系统的行波，可以对行波系统的相空间几何特征进行精细的分析和刻画，从而彻底搞清楚不可积系统中有界行波的产生机理，保持的条件和在分岔值附近的动力学行为。