On the structure of spectra of modulated travelling waves

Björn Sandstede, Arnd Scheel

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63 Scopus citations


Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses or fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the resolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appropriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floer index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.

Original languageEnglish (US)
Pages (from-to)39-93
Number of pages55
JournalMathematische Nachrichten
StatePublished - 2001


  • Exponential dichotomies
  • Modulated waves
  • Reaction-diffusion equations
  • Stability
  • Travelling waves


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