Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical “pattern-forming” instability and intertwine in a manner often referred to as “homoclinic snaking.” In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg–Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region of parameter space where localized states exist and are stable.