Singularly perturbed and non-local modulation equations for systems with interacting instability mechanisms

Abstract

Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilizing mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small { as for instance can be the case in double-layer convection. Based on these assumptions we rst derive a singularly perturbed modulation equation and then a modulation equation with a non-local term. The reduction of the singularly perturbed system to the non-local system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the non-local system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the `Landau-reduction' to the non-local system has no signicant in uence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called `localised structures' in the underlying system: they connect simple periodic patterns at x!1. None of these patterns can be described by the non-local system. So, one may conclude that the reduction to the non-local system destroys a rich and important set of patterns.