On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

J.A. Sherratt

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 5, page 64-79
  • ISSN: 0973-5348

Abstract

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

How to cite

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Sherratt, J.A.. "On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion." Mathematical Modelling of Natural Phenomena 5.5 (2010): 64-79. <http://eudml.org/doc/197714>.

@article{Sherratt2010,
abstract = {Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the particular equation ut = (uux)x + u(1 − u). Using singular perturbation theory, the author derives an asymptotic approximation to the wave, which gives valuable information about the structure of smooth-front solutions. The approximation compares well with numerical results.},
author = {Sherratt, J.A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {nonlinear diffusion; travelling waves; sharp front; smooth front; asymptotic approximation},
language = {eng},
month = {7},
number = {5},
pages = {64-79},
publisher = {EDP Sciences},
title = {On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion},
url = {http://eudml.org/doc/197714},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Sherratt, J.A.
TI - On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/7//
PB - EDP Sciences
VL - 5
IS - 5
SP - 64
EP - 79
AB - Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the particular equation ut = (uux)x + u(1 − u). Using singular perturbation theory, the author derives an asymptotic approximation to the wave, which gives valuable information about the structure of smooth-front solutions. The approximation compares well with numerical results.
LA - eng
KW - nonlinear diffusion; travelling waves; sharp front; smooth front; asymptotic approximation
UR - http://eudml.org/doc/197714
ER -

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