  In pattern-forming systems, localized patterns are readily found when stable
patterns exist at the same parameter values as the stable unpatterned state.
Oscillons are spatially localized, time-periodic structures, which have been
found experimentally in systems that are driven by a time-periodic force, for
example, in the Faraday wave experiment. This paper examines 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, arXiv:0805.0878] as a phenomenological model of
the Faraday wave experiment. We choose parameters so that patterns set in with
non-zero wavenumber (in contrast to [A. S. Alnahdi, J. Niesen and A. M.
Rucklidge, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1311-1327,
arXiv:1312.2773]). In the limit of weak damping, weak detuning, weak forcing,
small group velocity, and small amplitude, we reduce the model PDE to the
coupled forced complex Ginzburg-Landau equations. We find localized solutions
and snaking behaviour in the coupled forced complex Ginzburg-Landau equations
and relate these to oscillons that we find in the model PDE. Close to onset,
the agreement is excellent. The periodic forcing for the PDE and the explicit
derivation of the amplitude equations make our work relevant to the
experimentally observed oscillons.
