Mathematical Sciences: Traveling Wave Solutions of Reaction Diffusion Systems

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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.