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Mathematische Zeitschrift

, Volume 254, Issue 2, pp 359–383 | Cite as

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

  • Juncheng Wei
  • Matthias Winter
Article

Abstract

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

Open image in new window

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 Open image in new window approaches an optimal configuration in (*), where Open image in new window .

Keywords

FitzHugh-Nagumo system Standing waves Optimal configuration Localized energy method 

Mathematics Subject Classification

Primary 35B40 35B45 Secondary 35J55 92C15 92C40 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.School of Information Systems, Computing and MathematicsBrunel UniversityUxbridgeUnited Kingdom
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong Kong

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