Localized Patterns in Periodically Forced Systems

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [22] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg–Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [7]. We reduce the forced complex Ginzburg–Landau equation to the Allen–Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. We also extend this analysis to the strong forcing case recovering Allen–Cahn equation directly without the intermediate step. We find excellent agreement between numerical localized solutions of the PDE, localized solutions of the forced complex Ginzburg-Landau equation, and analytic (hyperbolic) solutions of the Allen–Cahn equation. This is the first time that a PDE with time dependent forcing has been reduced to the Allen–Cahn equation, and its localized oscillatory solutions quantitatively studied. This paper is dedicated to the memory of Thomas Wagenknecht.