Localized Patterns in Periodically Forced Systems: II. Patterns with Non-Zero Wavenumber

摘要

In pattern-forming systems, localized patterns are readily found when stablepatterns exist at the same parameter values as the stable unpatterned state.Oscillons are spatially localized, time-periodic structures, which have beenfound experimentally in systems that are driven by a time-periodic force, forexample, in the Faraday wave experiment. This paper examines the existence ofoscillatory localized states in a PDE model with single frequency timedependent 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 ofthe Faraday wave experiment. We choose parameters so that patterns set in withnon-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 thecoupled forced complex Ginzburg-Landau equations. We find localized solutionsand snaking behaviour in the coupled forced complex Ginzburg-Landau equationsand 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 explicitderivation of the amplitude equations make our work relevant to theexperimentally observed oscillons.