Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations
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In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(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 φ(x−c*t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation.
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