Mathematical Sciences: Dynamics and Patterns in Reaction Diffusion Equations

反应扩散方程可能会产生各种各样的 解决方案的模式。 其中包括行波 解决方案，稳定的解决方案，细胞模式， 振荡的解决方案，甚至解决方案，其中表现在一个 混乱的方式。 所有的系统， 研究者将考虑作为一个数学模型的出现， 生物或化学系统。 该项目将考虑 燃烧模型，催化表面上的化学反应， 相变和可兴奋膜的破裂。 的 该研究项目的主要目标是通过调查 这些模式可能产生的数学机制，分析 每个模式的稳定性属性，并了解 每种模式对潜在的 身体问题。 这个项目中要解决的问题提出了一些 基本的数学问题。 特别感兴趣的是 出现在不止一个空间维度上的模式。 大部分 分析将关注更高维度的模式如何 从基本上一维的物体如平面 波解和径向对称解。 简单 实例表明，分叉结构 模式可能非常复杂。

Reaction diffusion equations may give rise to a variety of pattern like solutions. These include traveling wave solutions, stable steady solutions, cellular patterns, oscillatory solutions, and even solutions which behave in a chaotic manner. All of the systems which the principle investigator will consider arise as a mathematical model of a biological or chemical system. This project will consider models for combusion, chemical reactions on catalytic surfaces, phase transitions, and bursting in excitable membranes. The primary goals of this research project are to investigate through which mathematical mechanisms these patterns may arise, analyse the stability properties of each pattern, and understand what physical significance each pattern has for the underlying physical problem. The problems to be addressed in this project raise a number of fundamental mathematical issues. Of particular interest are patterns arising in more than one space dimension. Much of the analysis will be concerned with how higher dimensional patterns bifurcate from basically one dimensional objects such as planar wave solutions and radially symmetric solutions. Simple examples demonstrate that the structure of the bifurcating patterns may be incredibly complex.