Hole solutions in the 1D complex Ginzburg-Landau equation

Stefan Popp, Olaf Stiller, Igor Aranson, Lorenz Kramer

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Abstract

The cubic Complex Ginzburg-Landau Equation (CGLE) has a one parameter family of traveling localized source solutions. These so called "Nozaki-Bekki holes" are (dynamically) stable in some parameter range, but always structurally unstable: A perturbation of the equation in general leads to a (positive or negative) monotonic acceleration or an oscillation of the holes. This confirms that the cubic CGLE has an inner symmetry. As a consequence small perturbations change some of the qualitative dynamics of the cubic CGLE and enhance or suppress spatio-temporal intermittency in some parameter range. An analytic stability analysis of holes in the cubic CGLE and a semianalytical treatment of the acceleration instability in the perturbed equation is performed by using matching and perturbation methods. Furthermore we treat the asymptotic hole-shock interaction. The results, which can be obtained fully analytically in the nonlinear Schrödinger limit, are also used for the quantitative description of modulated solutions made up of periodic arrangements of traveling holes and shocks.

Original languageEnglish (US)
Pages (from-to)398-423
Number of pages26
JournalPhysica D: Nonlinear Phenomena
Volume84
Issue number3-4
DOIs
StatePublished - Jul 1 1995

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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