Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations
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...
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| Format: | Journal article |
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1997
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| Summary: | 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. |
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