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Standing Pulse Solutions to FitzHugh–Nagumo Equations

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Abstract

Reaction–diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences. A localized pattern is a stable non-constant stationary solution usually located far away from neighborhoods of bifurcation induced by Turing’s instability. In the study of FitzHugh–Nagumo equations, we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system, and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses found in the literature, a new technique is attempted by seeking a standing pulse solution with a sign change.

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Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, Changhua, Taiwan, ROC

    Chao-Nien Chen

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Y. S. Choi

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  1. Chao-Nien Chen
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  2. Y. S. Choi
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Correspondence to Chao-Nien Chen.

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Communicated by P. Rabinowitz

Dedicated to Ta-Chung Lee on the occasion of his 65th birthday

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Chen, CN., Choi, Y.S. Standing Pulse Solutions to FitzHugh–Nagumo Equations. Arch Rational Mech Anal 206, 741–777 (2012). https://doi.org/10.1007/s00205-012-0542-3

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  • DOI: https://doi.org/10.1007/s00205-012-0542-3

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