# The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 4, page 269-292
- ISSN: 0973-5348

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topLatushkin, Y., and Sukhtayev, A.. "The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited." Mathematical Modelling of Natural Phenomena 5.4 (2010): 269-292. <http://eudml.org/doc/197679>.

@article{Latushkin2010,

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.},

author = {Latushkin, Y., Sukhtayev, A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Fredholm determinants; non-self-adjoint operators; Evans function; linear stability; traveling waves},

language = {eng},

month = {5},

number = {4},

pages = {269-292},

publisher = {EDP Sciences},

title = {The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited},

url = {http://eudml.org/doc/197679},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Latushkin, Y.

AU - Sukhtayev, A.

TI - The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/5//

PB - EDP Sciences

VL - 5

IS - 4

SP - 269

EP - 292

AB - 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.

LA - eng

KW - Fredholm determinants; non-self-adjoint operators; Evans function; linear stability; traveling waves

UR - http://eudml.org/doc/197679

ER -

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