The Mathematical Institute, University of Oxford, Eprints Archive

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

Sánchez-Garduño, F. and Maini, P. K. (1997) Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations. Journal of Mathematical Biology, 35 (6). pp. 713-728.

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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 $\phi(x-c^{*}t)$ is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for $c \neq c^{*}$. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation.

Item Type:Article
Uncontrolled Keywords:Sharp fronts · Degenerate diffusion · Hamiltonian · Bifurcation of heteroclinic trajectories
Subjects:A - C > Biology and other natural sciences
Research Groups:Centre for Mathematical Biology
ID Code:458
Deposited By:Philip Maini
Deposited On:06 Dec 2006
Last Modified:20 Jul 2009 14:21

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