Journal of Nonlinear Science

, Volume 26, Issue 5, pp 1369–1444 | Cite as

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

  • Paul CarterEmail author
  • Björn de Rijk
  • Björn Sandstede


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 \(\varepsilon \) 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 \(\varepsilon \), while the relevant scaling in the oscillatory case is \(\varepsilon ^{2/3}\).


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

Mathematics Subject Classification

35B35 35C07 35B25 35P15 35K57 



Carter was supported by the NSF under Grant DMS-1148284. De Rijk was supported by the Dutch science foundation (NWO) cluster NDNS+. Sandstede was partially supported by the NSF through Grant DMS-1409742.

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Paul Carter
    • 1
    Email author
  • Björn de Rijk
    • 2
  • Björn Sandstede
    • 3
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Mathematisch InstituutUniversiteit LeidenLeidenThe Netherlands
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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