Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

Abstract

In this paper we study the existence of one-dimensional travelling wave solutions for the non-linear degenerate (at u=0) reaction-diffusion equation 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 of c for which is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for . We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation.