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The Algebraic Multiplicity of Eigenvalues and the EvansFunction Revisited

Published online by Cambridge University Press:  12 May 2010

Y. Latushkin  and
A. Sukhtayev
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Abstract

This paper is related to the spectral stability of traveling wave solutions of partialdifferential equations. In the first part of the paper we use the Gohberg-Rouche Theoremto prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstractoperator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of thecorresponding Birman-Schwinger type operator pencil. In the second part of the paper weapply this result to discuss three particular classes of problems: the Schrödingeroperator, the operator obtained by linearizing a degenerate system of reaction diffusionequations about a pulse, and a general high order differential operator. We studyrelations between the algebraic multiplicity of an isolated eigenvalue for the respectiveoperators, and the order of the eigenvalue as the zero of the Evans function for thecorresponding first order system.

Keywords

Fredholm determinantsnon-self-adjoint operatorsEvans functionlinear stabilitytraveling waves
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 269 - 292
Copyright
© EDP Sciences, 2010

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Footnotes

Dedicated to the memory of M. S. Birman

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