Deformation of striped patterns by inhomogeneities

Gabriela Jaramillo, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We study the effects of adding a local perturbation in a pattern-forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction, which vanishes for a particular spatial shift only, which we interpret as a phase-selectionmechanism through the inhomogeneity.

Original languageEnglish (US)
Pages (from-to)51-65
Number of pages15
JournalMathematical Methods in the Applied Sciences
Volume38
Issue number1
DOIs
StatePublished - Jan 15 2015

Keywords

  • Ginzburg-Landau
  • Inhomogeneity

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