  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 [20] 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 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.
