The effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg–Landau equation

Type
Article

Authors
Aguareles, M.

KAUST Grant Number
KUK-C1-013-04

Date
2014-06

Abstract
In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg-Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg-Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q. © 2014 Elsevier B.V. All rights reserved.

Citation
Aguareles M (2014) The effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg–Landau equation. Physica D: Nonlinear Phenomena 278-279: 1–12. Available: http://dx.doi.org/10.1016/j.physd.2014.03.007.

Acknowledgements
The author thanks S.J. Chapman and T. Witelski for stimulating discussions. M. Aguareles has been supported in part by grants from the Spanish Government (MTM2011-27739-C04-03), from the Catalan Government (2009SGR345) and also by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The author would also like to thank the center OCIAM of the University of Oxford where part of this research was carried out.

Publisher
Elsevier BV

Journal
Physica D: Nonlinear Phenomena

DOI
10.1016/j.physd.2014.03.007

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