The stability and evolution of curved domains arising from one-dimensional localized patterns

Karl B Glasner, Alan E. Lindsay

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

In many pattern forming systems, narrow two-dimensional domains can arise whose cross sections are roughly one-dimensional localized solutions. This paper investigates this phenomenon in the variational Swift-Hohenberg equation. Stability of straight line solutions is analyzed, leading to criteria for either curve buckling or curve disintegration. Matched asymptotic expansions are used to derive a two-term expression for the geometric motion of curved domains, which includes both elastic and surface diffusion-type regularizations of curve motion. This leads to novel equilibrium curves and space-filling pattern proliferation. Numerical tests are used to confirm and illustrate these phenomena.

Original languageEnglish (US)
Pages (from-to)650-673
Number of pages24
JournalSIAM Journal on Applied Dynamical Systems
Volume12
Issue number2
DOIs
Publication statusPublished - 2013

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Keywords

  • Curvature
  • Geometric motion
  • Localized states
  • Matched asymptotics
  • Surface diffusion

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation

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