Computing absolute and essential spectra using continuation

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

Research output: Contribution to journalArticlepeer-review

80 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 languageEnglish (US)
Pages (from-to)166-183
Number of pages18
JournalPhysica D: Nonlinear Phenomena
Volume229
Issue number2
DOIs
StatePublished - 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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