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Patterns of sources and sinks in the complex ginzburg-landau equation with zero linear dispersion

Jonathan A. Sherratt, Matthew J. Smith, and Jens D. M. Rademacher


The complex Ginzburg-Landau equation with zero linear dispersion occurs in a wide variety of contexts as the modulation equation near the supercritical onset of a homogeneous oscillation. The analysis of its coherent structures is therefore of great interest. Its fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves and plane waves). In the past decade interfaces separating regions with different wavetrains have been studied in detail, as they occur both in simulations and in real experiments. The basic interface types are sources and sinks, distinguished by the signs of the opposing group velocities of the adjacent wavetrains. In this paper we study existence conditions for propagating patterns composed of sources and sinks. Our analysis is based on a formal asymptotic expansion in the limit of large source-sink separation and small speed of propagation. The main results concern the possible relative locations of sources and sinks in such a pattern. We show that sources and sinks are to leading order only coupled to their nearest neighbours, and that the separations of a source from its neighbouring sinks, L+ and L− say, satisfy a phase locking condition, whose leading order form is derived explicitly. Significantly this leading order phase locking condition is independent of the propagation speed. The solutions of the condition form a discrete infinite sequence of curves in the L+–L− plane.


Publication typeArticle
Published inSIAM Journal of Applied Dynamical Systems
PublisherSociety for Industrial and Applied Mathematics

Previous versions

Matthew J. Smith, Jens D. M. Rademacher, and Jonathan A. Sherratt. Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type, SIAM Journal of Applied Dynamical Systems, 8(3), 1136-1159, 27 August 2009.

Jonathan A. Sherratt, Matthew J. Smith, and Jens D.M. Rademacher. Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion, Proceedings of that National Academy of Sciences of the United States of America, 106(27), pp. 10890-10895, 7 July 2009.

Matthew Smith, Jonathan Sherratt, and Xavier Lambin. The effects of density dependent dispersal on the spatiotemporal dynamics of cyclic populations, Journal of Theoretical Biology, Elsevier, 21 September 2008.

Matthew J. Smith and Jonathan A. Sherratt. Propagating fronts in the complex Ginzburg-Landau equation generate fixed-width bands of plane waves., Physical Review E, 6 October 2009.

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