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Travelling Waves for Complete Discretizations of Reaction Diffusion Systems

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Abstract

In this paper we consider the impact that full spatial–temporal discretizations of reaction–diffusion systems have on the existence and uniqueness of travelling waves. In particular, we consider a standard second-difference spatial discretization of the Laplacian together with the six numerically stable backward differentiation formula methods for the temporal discretization. For small temporal time-steps and a fixed spatial grid-size, we establish some useful Fredholm properties for the operator that arises after linearizing the system around a travelling wave. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is associated to a lattice differential equation, where space has been discretized but time remains continuous. For the backward-Euler temporal discretization, we also obtain travelling waves for arbitrary time-steps. In addition, we show that in the anti-continuum limit, in which the temporal time-step and the spatial grid-size are both very large, wave speeds are no longer unique. This is in contrast to the situation for the original continuous system and its spatial semi-discretization. This non-uniqueness is also explored numerically and discussed extensively away from the anti-continuum limit.

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Acknowledgments

Hupkes acknowledges support from the Netherlands Organization for Scientific Research (NWO). Van Vleck acknowledges support from the NSF (DMS-1115408 and DMS-1419047).

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Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    H. J. Hupkes

  2. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS, 66045, USA

    E. S. Van Vleck

Authors
  1. H. J. Hupkes
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  2. E. S. Van Vleck
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Correspondence to H. J. Hupkes.

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Dedicated to Professor John Mallet-Paret on the occasion of his 60th birthday.

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Hupkes, H.J., Van Vleck, E.S. Travelling Waves for Complete Discretizations of Reaction Diffusion Systems. J Dyn Diff Equat 28, 955–1006 (2016). https://doi.org/10.1007/s10884-014-9423-9

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  • DOI: https://doi.org/10.1007/s10884-014-9423-9

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