A geometric classification of traveling front propagation in the Nagumo equation with cut-off

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Abstract

An important category of solutions to reaction-diffusion systems of partial differential equations is given by traveling fronts, which provide a monotonic connection between rest states and maintain a fixed profile when considered in a co-moving frame. Reaction-diffusion equations are frequently employed in the mean-field (continuum) approximation of discrete (many-particle) models; however, the quality of this approximation deteriorates when the number of particles is not sufficiently large. The (stochastic) effects of this discreteness have been modeled via the introduction of (deterministic) 'cut-offs' that effectively deactivate the reaction terms at points where the particle concentration is below a certain threshold. In this article, we present an overview of the effects of such a cut-off on the front propagation dynamics in a prototypical reaction-diffusion system, the classical Nagumo equation. Our analysis is based on the method of geometric desingularization ('blow-up'), in combination with dynamical systems techniques such as invariant manifolds and normal forms. Using these techniques, we categorize front propagation in the cut-off Nagumo equation in dependence of a control parameter, and we classify the corresponding propagation regimes ('pulled,' pushed,' and 'bistablel in terms of the bifurcation structure of a projectivized system of equations that is obtained from the original traveling front problem, after blow-up. In particular, our approach allows us to determine rigorously the asymptotics (in the cut-off parameter) of the correction to the front propagation speed in the Nagumo equation that is due to a cut-off. Moreover, it explains the structure of that asymptotics (logarithmic, superlinear, or sublinear) in dependence of the front propagation regime. Finally, it enables us to calculate the corresponding leading-order coefficients in the resulting expansions in closed form.

Original languageEnglish
Title of host publication5TH INTERNATIONAL WORKSHOP ON MULTI-RATE PROCESSES AND HYSTERESIS (MURPHYS 2010)
EditorsA Ivanyi, P Ivanyi, D Rachinskii, VA Sobolev
Place of PublicationBRISTOL
PublisherIOP Publishing Ltd.
Pages-
Number of pages26
ISBN (Print)*****************
DOIs
Publication statusPublished - 2010

Keywords

  • REACTION-DIFFUSION EQUATIONS
  • RIGOROUS ASYMPTOTIC EXPANSIONS
  • SINGULAR PERTURBATION-THEORY
  • CRITICAL WAVE SPEED
  • BLOW-UP
  • MODEL
  • FLUCTUATIONS
  • SELECTION
  • FAMILY

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