On traveling wave solutions of nonlinear diffusion equations

Abstract

An existence proof for periodic traveling wave solutions of an equation of Nagumo is outlined. The proof begins with the analysis of a limiting case in which one set of dependent variables moves infinitely fast compared to the remainder. Singular "orbits" are defined for this limiting system and a perturbation argument using isolating blocks allows one to find actual solutions. The problem discussed here concerns the existence of bounded solutions of a one parameter family of ordinary differential equations. The solutions correspond to traveling wave solutions of a diffusion equation; the parameter is the wave velocity. The approach here relies on being able to assume that some of the dependent variables change very rapidly compared to the others. One first considers a limiting case where the "slow" variables don't change at all; this has the effect that one studies a lower dimensional system which now depends on more parameters. Thus the limit case can be described as a flow on a fiber bundle the base space of which is the parameter (slow variable) space and the fiber of which is the fast variable space. Away from the rest points of this flow, the orbits in the fiber provide a good approximation to solutions of the actual equation .