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

3 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 (US)
Pages (from-to)739-786
Number of pages48
JournalNonlinearity
Volume27
Issue number4
DOIs
StatePublished - Apr 2014

Fingerprint

Complex Ginzburg-Landau Equation
Landau-Ginzburg equations
Nonlinear Stability
Defects
defects
Resolvent Estimates
Time-periodic Solutions
phase response
reaction-diffusion equations
Kernel Estimate
Pointwise Estimates
Periodic Wave
Burger equation
Iteration Scheme
Group Velocity
estimates
Burgers Equation
Far Field
Reaction-diffusion Equations
Green's function

All Science Journal Classification (ASJC) codes

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

Cite this

Beck, Margaret ; Nguyen, Toan T. ; Sandstede, Björn ; Zumbrun, Kevin. / Nonlinear stability of source defects in the complex Ginzburg-Landau equation. In: Nonlinearity. 2014 ; Vol. 27, No. 4. pp. 739-786.
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Nonlinear stability of source defects in the complex Ginzburg-Landau equation. / Beck, Margaret; Nguyen, Toan T.; Sandstede, Björn; Zumbrun, Kevin.

In: Nonlinearity, Vol. 27, No. 4, 04.2014, p. 739-786.

Research output: Contribution to journalArticle

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