Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Thiele, Uwe; Archer, Andrew J.; Robbins, Mark J.; Gómez, Héctor; Knobloch, Edgar

doi:10.1103/physreve.87.042915

Cited by 93 publications

(195 citation statements)

References 84 publications

(157 reference statements)

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“…It is also known that the form of the resulting bifurcation diagram is significantly altered in the presence of a conserved quantity. 15,16 This is because the localized structures typically expel the conserved quantity thereby raising its magnitude in the region outside. This in turn modifies the background state and leads to so-called slanted snaking.…”

Section: Introductionmentioning

confidence: 99%

“…This in turn modifies the background state and leads to so-called slanted snaking. [15][16][17][18] The presence of slanted snaking implies that localized states are present over a much wider interval in parameter space than is the case with standard snaking. Slanted snaking is a consequence of a conserved quantity, such as imposed magnetic flux in magnetoconvection 19 or fixed zonal momentum in rotating convection with stress-free boundary conditions at top and bottom, 18 and is a finite size effect -in an unbounded domain the conserved quantity exerts no effect and the system reverts to standard snaking.…”

Section: Introductionmentioning

confidence: 99%

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Localized rotating convection with no-slip boundary conditions

et al. 2013

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Localized patches of stationary convection embedded in a background conduction state are called convectons. Multiple states of this type have recently been found in two-dimensional Boussinesq convection in a horizontal fluid layer with stress-free boundary conditions at top and bottom, and rotating about the vertical. The convectons differ in their lengths and in the strength of the self-generated shear within which they are embedded, and exhibit slanted snaking. We use homotopic continuation of the boundary conditions to show that similar structures exist in the presence of no-slip boundary conditions at the top and bottom of the layer and show that such structures exhibit standard snaking. The homotopic continuation allows us to study the transformation from slanted snaking characteristic of systems with a conserved quantity, here the zonal momentum, to standard snaking characteristic of systems with no conserved quantity. C 2013 AIP Publishing LLC. [http://dx

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Localized rotating convection with no-slip boundary conditions

et al. 2013

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“…However, a similar bifurcation structure is found for the classical conserved Swift-Hohenberg equation, that is, the regular PFC model. We will present results from our study of localized states in the regular PFC model elsewhere [27].…”

Section: Equation (13) Then Yieldsmentioning

confidence: 99%

Modeling the structure of liquids and crystals using one- and two-component modified phase-field crystal models

et al. 2012

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A modified phase field crystal model in which the free energy may be minimised by an order parameter profile having isolated bumps is investigated.The phase diagram is calculated in one and two dimensions and we locate the regions where modulated and uniform phases are formed and also regions where localised states are formed. We investigate the effectiveness of the phase field crystal model for describing fluids and crystals with defects. We further consider a two component model and elucidate how the structure transforms from hexagonal crystalline ordering to square ordering as the concentration changes.Our conclusion contains a discussion of possible interpretations of the order parameter field.

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“…Dissipative localized structures have been theoretically predicted and experimentally observed in various fields of natural science such as biology, chemistry, ecology, physics, fluid mechanics, and optics (see e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). Localized structures of light in the transverse section of passive and active optical devices are often called cavity solitons.…”

Section: Introductionmentioning

confidence: 99%

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

et al. 2017

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We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial gaussian pumping beam, and subjected to timedelayed feedback. The gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assumming a continous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use two approaches based on either an order parameter equation, describing the dynamics of the localized structure in the vicinity of the Hopf bifurcation or an overdamped dynamics of a particle in a potential well generated by the inhomogeneity. In the later approach, the time-delayed feedback acts as a driving force.

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Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Cited by 93 publications

References 84 publications

Localized rotating convection with no-slip boundary conditions

Localized rotating convection with no-slip boundary conditions

Modeling the structure of liquids and crystals using one- and two-component modified phase-field crystal models

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

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