Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

Abstract

In this paper we study the existence of one-dimensional travelling wave solutions u(x, t) = phi(x - ct) for the non-linear degenerate (at u = 0) reaction-diffusion equation u(t) = [D(u)u(x)](x) + g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c = 0, 2. The existence of a unique value c > 0 of c for which phi(x - ct) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c not equal c. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation.