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smooth front

Mathematical Modelling of Natural Phenomena (1)

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On the Form of Smooth-Front Travelling Waves in aReaction-Diffusion Equation with Degenerate Nonlinear Diffusion
J.A. Sherratt
Journal:

Mathematical Modelling of Natural Phenomena / Volume 5 / Issue 5 / 2010

Published online by Cambridge University Press:

27 July 2010, pp. 64-79

Print publication:

2010
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Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use asmodels of biological phenomena. This paper begins with a survey of applications toecology, cell biology and bacterial colony patterns. The author then reviews mathematicalresults on the existence of travelling wave front solutions of these equations, and theirgeneration from given initial data. A detailed study is then presented of the form ofsmooth-front waves with speeds close to that of the (unique) sharp-front solution, for theparticular equation ut =(uux)x + u(1− u). Using singular perturbation theory, the author derives anasymptotic approximation to the wave, which gives valuable information about the structureof smooth-front solutions. The approximation compares well with numerical results.