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Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry

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Abstract

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system

where 0<a<1/2 and δ γ=β 2 ∈ (0,a), and the following combinatorial problem:

(*) Given K points Q 1 , . . . , Q K ∈R N with minimum distance 1, find out the maximum number of times that the minimum distance 1 can occur.

More precisely, we show that for any given positive integer K, there is a δ K >0 such that for 0<δ<δ K , there exists a standing-wave solution (u δ ,ν δ ) to the FitzHugh-Nagumo system with the property that u δ has K spikes Q δ 1,. . .,Q δ K and approaches an optimal configuration in (*), where .

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Authors and Affiliations

  1. School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, UB8 3PH, United Kingdom

    Juncheng Wei

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Matthias Winter

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  1. Juncheng Wei
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  2. Matthias Winter
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Correspondence to Matthias Winter.

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Wei, J., Winter, M. Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry. Math. Z. 254, 359–383 (2006). https://doi.org/10.1007/s00209-006-0952-8

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  • DOI: https://doi.org/10.1007/s00209-006-0952-8

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