Pulse replication and accumulation of eigenvalues

Paul Carter, Jens D.M. Rademacher, Bjorn Sandstede

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Motivated by pulse-replication phenomena observed in the FitzHugh-Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that the spectra of the PDE linearization about such pulses may contain many point eigenvalues that accumulate onto a union of curves as the slow scale parameter approaches zero. The limit sets are related to the absolute spectrum of the homogeneous rest states involved in the canard-like transitions. Our results are formulated for general systems that admit an appropriate slow/fast structure.

Original languageEnglish (US)
Pages (from-to)3520-3576
Number of pages57
JournalSIAM Journal on Mathematical Analysis
Volume53
Issue number3
DOIs
StatePublished - 2021

Bibliographical note

Funding Information:
\ast Received by the editors May 26, 2020; accepted for publication (in revised form) April 26, 2021; published electronically June 21, 2021. https://doi.org/10.1137/20M1340113 Funding: The work of the first author was supported by National Science Foundation grant DMS-2016216 (formerly DMS-1815315). The work of the second author was supported by the German Research Fund (DFG) through grant RA 2788/1-1. The work of the third author was supported by National Science Foundation grant DMS-1714429. \dagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (pcarter@ umn.edu). \ddagger Fachbereich Mathematik, University of Bremen, 28359, Bremen, Germany (jdmr@uni-bremen.de). \S Division of Applied Mathematics, Brown University, Providence, RI 02912 USA (Bjorn Sandstede@brown.edu).

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.

Keywords

  • Absolute spectrum
  • Canards
  • FitzHugh-Nagumo equation
  • Geometric singular perturbation theory
  • Spectral stability
  • Traveling pulses

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