Swift-Hohenberg model for magnetoconvection

Cox, Stephen M.; Matthews, P. C.; Sl, Pollicott

doi:10.1103/physreve.69.066314

Cited by 13 publications

(27 citation statements)

References 27 publications

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“…There are close connections between this work and that of Matthews and Cox and others [24,9,10,16,36]; these authors studied systems essentially equivalent to (2.1)-(2.2) obtained by applying ∂ 2 xx to the right-hand side of the standard Swift-Hohenberg equation (1.1), including both quadratic and cubic nonlinearities in N (w). In these papers the weakly nonlinear analysis proceeds by looking for small distortions of the large-scale mode, i.e., a small parameter δ and a long lengthscale X = δx are introduced, before expanding w = δa(X) sin x + O(δ 2 ) and B = 1 + δ 2 b(X) + O(δ 3 ).…”

Section: Discussionmentioning

confidence: 74%

“…While asymptotically correct, these scalings are unable to capture solutions where the fluctuations in the large-scale mode are large. As a result, numerical solutions often blow up, and higher-order stabilizing terms were found to be required [10]. Similar difficulties were noted by Golovin, Davis, and Voorhees [16].…”

mentioning

confidence: 86%

“…Our broad motivation comes from three physical situations: thermal convection in the presence of a vertical magnetic field [13,10], vertically vibrated granular media [17,34], and thin films [16]. In all three cases the large-scale mode arises due to the existence of a conserved quantity in the dynamics.…”

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confidence: 99%

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Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Dawes¹

2008

SIAM J. Appl. Dyn. Syst.

114

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Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical "pattern-forming" instability and intertwine in a manner often referred to as "homoclinic snaking." In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg-Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region of parameter space where localized states exist and are stable.

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Section: Discussionmentioning

confidence: 74%

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confidence: 86%

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Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Dawes¹

2008

SIAM J. Appl. Dyn. Syst.

114

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“…Existing attempts to explain the existence of convectons in these equations use matched asymptotic expansions valid in the limit of small diffusivity ratio ζ (Dawes 2007) or model equations in the spirit of the Swift-Hohenberg equation (Cox, Matthews & Pollicott 2004;Dawes 2008). In fact, the magnetoconvection problem differs in an important way from other systems exhibiting convectons.…”

Section: Introductionmentioning

confidence: 97%

Magnetohydrodynamic convectons

2011

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Numerical continuation is used to compute branches of spatially localized structures in convection in an imposed vertical magnetic field. In periodic domains with finite spatial period, these branches exhibit slanted snaking and consist of localized states of even and odd parity. The properties of these states are analysed and related to existing asymptotic approaches valid either at small amplitude (Cox and Matthews, Physica D, vol. 149, 2001, p. 210), or in the limit of small magnetic diffusivity (Dawes, J. Fluid Mech., vol. 570, 2007, p. 385). The transition to standard snaking with increasing domain size is explored.

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“…However, this mode does not adequately explain this phenomenon. Then some generalizations of this model have been derived in various branches in [8][9][10][11][12], such as nonlinear optics for lasers, magnetoconvection, and biological, chemical, and liquidcrystal light-valve experiment. In addition, the complex Swift-Hohenberg equation has been derived in [13,14].…”

Section: Introductionmentioning

confidence: 99%

Existence and Approximation of Manifolds for the Swift-Hohenberg Equation with a Parameter

Guo

2018

Discrete Dynamics in Nature and Society

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The existence and approximation of manifolds for the Swift-Hohenberg equation with a proper parameter have mainly been studied. Using the backward-forward systems from Swift-Hohenberg equation, the existence and specific representation forms of manifolds for Swift-Hohenberg equation with a parameter have been obtained. Meanwhile, we make use of technique of deposition of lower and higher frequency spaces of solutions and assume the reduced system to obtain the main numeration approximation system of approximation solution for the original system Swift-Hohenberg equation with a proper parameter.

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Swift-Hohenberg model for magnetoconvection

Cited by 13 publications

References 27 publications

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Magnetohydrodynamic convectons

Existence and Approximation of Manifolds for the Swift-Hohenberg Equation with a Parameter

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