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On the Structure of Spectra of Modulated Travelling Waves
Author(s) -
Sandstede Björn,
Scheel Arnd
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200112)232:1<39::aid-mana39>3.0.co;2-5
Subject(s) - mathematics , mathematical analysis , linearization , eigenvalues and eigenvectors , fredholm operator , elliptic curve , reaction–diffusion system , nonlinear system , physics , compact operator , quantum mechanics , computer science , extension (predicate logic) , programming language
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 o 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 esolvent 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 appopriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse‐Floe 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.