Patterns of Sources and Sinks in the Complex Ginzburg–Landau Equation with Zero Linear Dispersion

Abstract

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 coupled only to their nearest neighbors, and that the separations of a source from its neighboring 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.