Evans function and blow-up methods in critical eigenvalue problems

Björn Sandstede, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

Contact defects are one of several types of defects that arise generically in oscillatory media modelled by reaction-diffusion systems. An interesting property of these defects is that the asymptotic spatial wavenumber is approached only with algebraic order O(1/x) (the associated phase diverges logarithmically). The essential spectrum of the PDE linearization about a contact defect always has a branch point at the origin. We show that the Evans function can be extended across this branch point and discuss the smoothness properties of the extension. The construction utilizes blow-up techniques and is quite general in nature. We also comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons.

Original languageEnglish (US)
Pages (from-to)941-964
Number of pages24
JournalDiscrete and Continuous Dynamical Systems
Volume10
Issue number4
DOIs
StatePublished - Jun 2004

Keywords

  • Algebraic decay
  • Evans function
  • Radial Laplacian

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