TY - JOUR

T1 - Hole solutions in the 1D complex Ginzburg-Landau equation

AU - Popp, Stefan

AU - Stiller, Olaf

AU - Aranson, Igor

AU - Kramer, Lorenz

N1 - Funding Information:
We wish to thank S. Sasa for sending us data of the core instability line in addition to those in \[9\]. One of us (I.A.) wishes to thank the Alexander-von-Humboldt Stiftung for financial support and the University of Bayreuth for its hospitality. Support by the Deutsche Forschungsgemeinschaft (Kr-690/4, Schwerpunkt 'Strukturbildung in dissipativen kontinuierlichen Systemen: Experiment und Theorie im quantitativen Vergleich' ) and the German-Israeli Foundation (GIF) is gratefully acknowledged.

PY - 1995/7/1

Y1 - 1995/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=58149209842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149209842&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(95)00070-K

DO - 10.1016/0167-2789(95)00070-K

M3 - Article

AN - SCOPUS:58149209842

VL - 84

SP - 398

EP - 423

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -