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

点击翻译按钮获取中文摘要

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.