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International Conference on Difference Equations and Applications

ICDEA 2018: Difference Equations and Discrete Dynamical Systems with Applications pp 55–112Cite as

Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 312))

Abstract

We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh–Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.

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Notes

  1. 1.

    Actually, in order to ensure that the boundary conditions (23) are satisfied one needs to consider perturbations \(v \in W^{1,p}\) for \(1 \le p < \infty \) while taking \(\varPhi \in W^{1, \infty }\).

  2. 2.

    Here we use modulo arithmetic on i.

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Acknowledgements

HJH, LM and WSS acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grants 639.032.612 and 613.001.304). EVV was supported in part by NSF grant # DMS-1714195.

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Authors and Affiliations

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

    Hermen Jan Hupkes, Leonardo Morelli & Willem M. Schouten-Straatman

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

    Erik S. Van Vleck

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  1. Hermen Jan Hupkes
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  4. Erik S. Van Vleck
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  1. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    Martin Bohner

  2. Institute for Analysis, CfD, Dresden University of Technology, Dresden, Germany

    Stefan Siegmund

  3. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Republic

    Roman Šimon Hilscher

  4. Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

    Petr Stehlík

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Hupkes, H.J., Morelli, L., Schouten-Straatman, W.M., Van Vleck, E.S. (2020). Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations. In: Bohner, M., Siegmund, S., Šimon Hilscher, R., Stehlík, P. (eds) Difference Equations and Discrete Dynamical Systems with Applications. ICDEA 2018. Springer Proceedings in Mathematics & Statistics, vol 312. Springer, Cham. https://doi.org/10.1007/978-3-030-35502-9_3

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