Pushed and pulled fronts in a discrete reaction–diffusion equation

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• 作者：J. R. King ; R. D. O’Dea
• 关键词：Discrete reaction–diffusion equation ; Liouville–Green ; Matched ; asymptotic analysis ; Travelling waves
• 刊名：Journal of Engineering Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：102
• 期：1
• 页码：89-116
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Classical Mechanics; Applications of Mathematics; Analysis; Mathematical Modeling and Industrial Mathematics;
• 出版者：Springer Netherlands
• ISSN：1573-2703
• 卷排序：102

文摘

We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction–diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength \(\mu \) between lattice points and on a detuning parameter (a) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of ‘pulled’ fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of ‘pushed’ fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of \(\mu \) and a.