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Mathematics > Analysis of PDEs

arXiv:1804.01727 (math)
[Submitted on 5 Apr 2018 (v1), last revised 17 Sep 2021 (this version, v3)]

Title:Multiple front standing waves in the FitzHugh-Nagumo equations

Authors:Chao-Nien Chen (NTHU), Eric Séré (CEREMADE)
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Abstract:There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite quadratic form in the velocity terms. When the system has two stable hyperbolic equilibria, there exist two stable standing fronts, which will be used in this paper as building blocks, to construct stable standing waves with multiple fronts in case the equilibria are of saddle-focus type. The idea to prove existence is somewhat close in spirit to [Buffoni-Sere, CPAM 49, 285-305]. However several differences are required in the argument: facing a strongly indefinite functional, we need to perform a nonlo-cal Lyapunov-Schmidt reduction; in order to justify the stability of multiple front standing waves, we rely on a more precise variational characterization of such critical points. Based on this approach, both stable and unstable standing waves are established.
Comments: Final version to appear in Journal of Differential Equations
Subjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Pattern Formation and Solitons (nlin.PS)
Cite as: arXiv:1804.01727 [math.AP]
  (or arXiv:1804.01727v3 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1804.01727
Journal reference: Journal of Differential Equations 302 (2021), pp. 895-925
Related DOI: https://doi.org/10.1016/j.jde.2021.08.005

Submission history

From: Eric Sere [view email] [via CCSD proxy]
[v1] Thu, 5 Apr 2018 08:26:44 UTC (29 KB)
[v2] Tue, 23 Apr 2019 13:13:13 UTC (44 KB)
[v3] Fri, 17 Sep 2021 10:00:49 UTC (36 KB)
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