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Oscillons: localized patterns in a periodically forced system

Alnahdi, Abeer Saleh (2015) Oscillons: localized patterns in a periodically forced system. PhD thesis, University of Leeds.

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Abstract

Spatially localized, time-periodic structures, known as oscillons, are common in patternforming systems, appearing in fluid mechanics, chemical reactions, optics and granular media. This thesis examines the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [70] as phenomenological model of the Faraday wave experiment. Firstly in the case where the prefered wavenumber at onset is zero, 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 [15]. We reduce the forced complex Ginzburg– Landau equation to the Allen–Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. In the strong forcing case, we get the Allen–Cahn equation directly. Throughout, we use continuation techniques to compute numerical solutions of the PDE model and the reduced amplitude equation. We do quantitative comparison of localized solutions and bifurcation diagrams between the PDE model, the forced complex Ginzburg–Landau equation, and the Allen–Cahn equation. The second aspect in this work concerns the investigation of the existence of localized oscillons that arise with non-zero preferred wavenumber. In the limit of weak damping, weak detuning, weak forcing, small group velocity, and small amplitude, asymptotic reduction of the model PDE to the coupled forced complex Ginzburg–Landau equations is done. In the further limit of being very close to onset, we reduce the coupled forced complex Ginzburg–Landau equations to the real Ginzburg–Landau equation. We have qualitative prediction of finding exact localized solutions from the real Ginzburg–Landau equation limited by computational constraints of domain size. Finally, we examine the existence of localized oscillons in the PDE model with cubic–quintic nonlinearity in the strong damping, strong forcing and large amplitude case. We find two snaking branches in the bistability region between stable periodic patterns and the stable trivial state in one spatial dimension in a manner similar to systems without time dependent forcing. We present numerical examples of localized oscillatory spots and rings in two spatial dimensions.

Item Type: Thesis (PhD)
Keywords: pattern formation, oscillons, localized states, forced complex Ginzburg–Landau equation.
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.656999
Depositing User: Mrs Abeer Alnahdi
Date Deposited: 21 Jul 2015 14:00
Last Modified: 25 Nov 2015 13:48
URI: http://etheses.whiterose.ac.uk/id/eprint/9422

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