Mathematical Sciences: Dynamical Systems Arising in the Stability Analysis of Traveling Waves

该奖项支持对行波稳定性的持续研究。一些应用领域，如流体力学(修正的Burgers方程)、反应扩散系统、模式分叉的Murray-Oster机械力化学理论和竞争种群模型。关于这些方程的波解已经做了大量的工作。这个项目关注的是这种波解的稳定性，这是一个很少受到关注的话题。利用拓扑不变量为行波稳定性分析提供了一个非常重要的框架。这是一个概念上的新概念；由于它的拓扑特性，对扰动非常健壮。“稳定指数”的基本概念是最近发展起来的。它作为一种新的工具显示出巨大的潜力，可以用来探索理论数学和物理世界中发生的许多波动现象的例子。行波是用于理解激波、晶体生长和火焰传播等动态图案形成的最基本对象，因为相变只通过界面边界的运动发生，并且这种运动采取行波的形式。重要的问题是知道这样的波是否可观察到，这与询问它们是否稳定是一样的。一般情况下，很难确定它们的稳定性，因为行波是连接两个不同稳定状态的大振幅解，并且通常包含尖锐的过渡层。本项目中用来检测稳定性的新的拓扑机制不是摄动机制，因此具有广泛的适用性。

This award supports continuing research on the stability of travelling waves. Several application areas such as fluid mechanics (a modified Burgers equation), reaction-diffusion systems, the Murray-Oster mechanochemical theory of pattern bifurcation and competing population models. Much work has been done on wave solutions of most of these equations. This project is concerned with the stability of such wave solutions, a topic which has received very little attention. The project gives a very important framework of stability analysis for travelling waves by using topological invariants. It is a conceptually new idea; very robust to perturbations because of its topological character. The basic idea of 'stability index' has been developed recently. It shows great potential as a new tool which can be used to explore many examples of wave phenomena occurring in both theoretical mathematics and the physical world. The travelling wave is the most basic object used to understand dynamic pattern formation such as shock waves, crystal growth, and flame propagation, since the phase change occurs only through the motion of interfacial boundaries and such a motion takes the form of travelling waves. The important issue is knowing whether such waves are observable or not, which is the same thing as asking whether they are stable. It is, in general, quite difficult to determine their stability properties, since travelling waves are large amplitude solutions connecting two different stable states and usually contain sharp transition layers. The new topological machinery used in this project to detect stability is not a perturbative one, and hence has the potential for broad applicability.