Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

Paul Carter, Björn de Rijk, Björn Sandstede

Research output: Contribution to journalArticle

12 Scopus citations


The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter ε goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with ε, while the relevant scaling in the oscillatory case is ε2 / 3.

Original languageEnglish (US)
Pages (from-to)1369-1444
Number of pages76
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2016
Externally publishedYes


  • FitzHugh-Nagumo system
  • Lin’s method
  • geometric singular perturbation theory
  • spectral stability
  • traveling pulses

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