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Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

By F. Sánchez-Garduño and P. K. Maini

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

Topics: Biology and other natural sciences
Year: 1997
DOI identifier: 10.1007/s002850050073
OAI identifier: oai:generic.eprints.org:458/core69

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Citations

  1. Nonlinear Diffusion in population, genetics and nerve pulse propagation, in Partial Differential Equations and Related Topics,
  2. (1973). Theory of Dynamic Systems on a Plane (Israel Program for Scientific Translations, doi

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