Alnahdi, AS, Niesen, J orcid.org/0000-0002-6693-3810 and Rucklidge, AM orcid.org/0000-0003-2985-0976 (2018) Localized Patterns in Periodically Forced Systems: II. Patterns with Non-Zero Wavenumber. SIAM Journal on Applied Dynamical Systems, 17 (2). pp. 1478-1502. ISSN 1536-0040
Abstract
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. Dyn. Syst., 8 (2009), pp. 298--347] 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]). 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 behavior 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | (c) 2018, Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Apr 2018 08:23 |
Last Modified: | 25 May 2018 15:44 |
Status: | Published |
Publisher: | Society of Industrial and Applied Mathematics |
Identification Number: | 10.1137/17M1162330 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:129439 |