Nonlinear stability of source defects in the complex Ginzburg-Landau equation

Margaret Beck, Toan T. Nguyen, Björn Sandstede, Kevin Zumbrun

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for Green's function, which allow one to close a nonlinear iteration scheme.

Original languageEnglish
Pages (from-to)739-786
Number of pages48
JournalNonlinearity
Volume27
Issue number4
DOIs
Publication statusPublished - 2014

Keywords

  • Burgers equation
  • defects
  • Ginzburg-Landau equation
  • Green function
  • nonlinear stability
  • patterns
  • sources

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

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