Computing absolute and essential spectra using continuation

Jens D.M. Rademacher; Björn Sandstede; Arnd Scheel

doi:10.1016/j.physd.2007.03.016

Computing absolute and essential spectra using continuation

Jens D.M. Rademacher, Björn Sandstede, Arnd Scheel

School of Mathematics

Research output: Contribution to journal › Article › peer-review

93 Scopus citations

Abstract

A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantages of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package auto and provide example computations for the FitzHugh-Nagumo and the complex Ginzburg-Landau equation.

Original language	English (US)
Pages (from-to)	166-183
Number of pages	18
Journal	Physica D: Nonlinear Phenomena
Volume	229
Issue number	2
DOIs	https://doi.org/10.1016/j.physd.2007.03.016
State	Published - May 15 2007

Bibliographical note

Funding Information:
J. Rademacher acknowledges hospitality at the Free University of Berlin and support from NSF grant DMS-0203301, a PIMS fellowship and DFG priority program SPP 1095. B. Sandstede acknowledges support from the NSF through grant DMS-0203854 and from a Royal Society Wolfson Research Merit Award. A. Scheel was partially supported by the NSF through grants DMS-0203301 and DMS-0504271.

Keywords

Absolute spectrum
Continuation
Instability thresholds
Reaction-diffusion systems
Spectral stability

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10.1016/j.physd.2007.03.016

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title = "Computing absolute and essential spectra using continuation",

abstract = "A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantages of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package auto and provide example computations for the FitzHugh-Nagumo and the complex Ginzburg-Landau equation.",

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note = "Funding Information: J. Rademacher acknowledges hospitality at the Free University of Berlin and support from NSF grant DMS-0203301, a PIMS fellowship and DFG priority program SPP 1095. B. Sandstede acknowledges support from the NSF through grant DMS-0203854 and from a Royal Society Wolfson Research Merit Award. A. Scheel was partially supported by the NSF through grants DMS-0203301 and DMS-0504271.",

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AU - Sandstede, Björn

AU - Scheel, Arnd

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N2 - A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantages of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package auto and provide example computations for the FitzHugh-Nagumo and the complex Ginzburg-Landau equation.

AB - A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantages of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package auto and provide example computations for the FitzHugh-Nagumo and the complex Ginzburg-Landau equation.

KW - Absolute spectrum

KW - Continuation

KW - Instability thresholds

KW - Reaction-diffusion systems

KW - Spectral stability

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Computing absolute and essential spectra using continuation

Abstract

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