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The Algebraic Multiplicity of Eigenvalues and the Evans Function 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 partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schrödinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding 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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