Mathematical Sciences: Traveling Wave Solutions of Reaction Diffusion Systems

反应扩散方程作为模型出现在数学、生物和化学的许多领域。其中许多系统的一个共同特征是存在行波解。它们对应于似乎以恒定的形状和速度行进的解。这个项目的目标是发展新的技术来证明行波解的存在性和稳定性。在本项目中概述的计划中有两个基本步骤。第一部分是发展证明行波解的存在性和稳定性的一般方法。这些方法的一个基本组成部分是康利指数，这是已故的查尔斯·康利教授在1977-1978年间开发的一种相对较新的工具。第二部分是基本理论的应用。要考虑的具体应用是生态和燃烧产生的行波解的存在性和稳定性，以及半线性椭圆组的径向对称解的存在性。最后一个问题与行波解的存在性问题密切相关。特曼教授提出的这个项目包含几个子项目，与这个数学理论与其他科学问题的各种潜在联系有关。特曼教授是一位年轻的有才华的数学家，他已经在这一研究领域确立了领先地位。

Reaction-diffusion equations arise as models in many areas of Mathematical Biology and Chemistry. A feature common to many of these systems is the existence of traveling wave solutions. These correspond to solutions which appear to be traveling with constant shape and velocity. The objective of this project is to develop new techniques for proving the existence and stability of traveling wave solutions. There are two basic steps in the program outlined in this project. The first consists of developing general methods for proving the existence and stability of traveling wave solutions. A basic ingredient for these methods is the Conley index, a relatively recent tool developed by the late Professor Charles Conley in 1977-1978. The second part consists of applications of the basic theory. Specific applications to be considered are the existence and stability of traveling wave solutions arising from ecology and combustion, and the existence of radially symmetric solutions of semilinear elliptic systems. This last problem is closely related to the question of the existence of traveling wave solutions. The project presented by Professor Terman contains several sub- projects related to various potential connections of this mathematical theory to other scientific problems. Professor Terman is a young talented mathematician who has already established his leading position in this area of research.