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 [A. M. Rucklidge and M. Silber, SIAM J. Appl. Math., 8 (2009), pp. 298--347] as a 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 [J. Burke, A. Yochelis, and E. Knobloch, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 651--711]. 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 the 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.