Pattern Formation

Hoyle, Rebecca B.

doi:10.1017/cbo9780511616051

Cited by 340 publications

(63 citation statements)

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“…4(a) and 4(b) show the post-instability density patterns forg = 12, β = 0.28 and α = 1.35 radians at times t = 13.5/ω z and t = 27/ω z , respectively. Note that the the first stages of the post-instability dynamics are characterized by the formation of a transient stripe pattern that present dislocation defects [36]. At these dislocations two stripes merge into one.…”

Section: Soliton Gas Formation After Phonon Instabilitymentioning

confidence: 99%

Two-dimensional bright solitons in dipolar Bose-Einstein condensates with tilted dipoles

et al. 2015

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The effect of dipolar orientation with respect to the soliton plane on the physics of two-dimensional bright solitons in dipolar Bose-Einstein condensates is discussed. Previous studies on such a soliton involved dipoles either perpendicular or parallel to the condensate-plane. The tilting angle constitutes an additional tuning parameter, which help us to control the in-plane anisotropy of the soliton as well as provides access to previously disregarded regimes of interaction parameters for soliton stability. In addition, it can be used to drive the condensate into phonon instability without changing its interaction parameters or trap geometry. The phonon-instability in a homogeneous 2D condensate of tilted dipoles always features a transient stripe pattern, which eventually breaks into a metastable soliton gas. Finally, we demonstrate how a dipolar BEC in a shallow trap can eventually be turned into a self-trapped matter wave by an adiabatic approach, involving the tuning of tilting angle.

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Section: Soliton Gas Formation After Phonon Instabilitymentioning

confidence: 99%

Two-dimensional bright solitons in dipolar Bose-Einstein condensates with tilted dipoles

et al. 2015

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“…This equation describes pattern-forming systems near instabilities with nontrivial finite spatial wavelength [15,21] but has variational dynamics with energy Local minima of E correspond to stable stationary solutions of (1.1). While the physical examples motivating our work are in general non-variational, it turns out that the Swift-Hohenberg equation sheds a great deal of light on the growth of stationary spatially localised structures in general.…”

Section: Introductionmentioning

confidence: 99%

To Snake or Not to Snake in the Planar Swift–Hohenberg Equation

Avitabile¹,

Lloyd²,

Burke³

et al. 2010

SIAM J. Appl. Dyn. Syst.

100

135

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We investigate the bifurcation structure of stationary localised patterns of the two-dimensional SwiftHohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localised roll, square and stripe patches that exhibit snaking and non-snaking behaviour on the same bifurcation branch. Some of these patterns snake between four saddle-node limits: recent analytical results predict then the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localised roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena we encounter.

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“…The Swift-Hohenberg equation is a general model for pattern-forming processes which was first derived by Swift and Hohenberg [39] to describe random thermal fluctuations in the Boussinesq equation. Equation (1.1) exhibits many interesting localized and non-localized patterns, and we refer to [10,14,17,29,30] for details; see also Figure 1 and §9. In particular, localized radial pulses and localized hexagon and rhomboid patches have been found in numerical investigations of the Swift-Hohenberg equation.…”

Section: Introductionmentioning

confidence: 99%

Localized radial solutions of the Swift–Hohenberg equation

Lloyd

Sandstede²

2009

Nonlinearity

150

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Stationary localized solutions of the planar Swift-Hohenberg equation are investigated in the parameter region where the trivial solution is stable. In the parameter region where rolls bifurcate subcritically, localized radial ring-like pulses are shown to bifurcate from the trivial solution. Furthermore, radial spot-like pulses are shown to bifurcate from the trivial state, regardless of the criticality of roll patterns. These theoretical results apply also to general reaction-diffusion systems near Turing instabilities. Numerical computations show that planar radial pulses 'snake' near the Maxwell point where, by definition, the one-dimensional roll patterns have the same energy as the trivial state. These computations also reveal that spots, which are stable in a certain parameter region, become unstable with respect to hexagonal perturbations, leading to fully localized hexagon patterns.

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Pattern Formation

Cited by 340 publications

References 0 publications

Two-dimensional bright solitons in dipolar Bose-Einstein condensates with tilted dipoles

Two-dimensional bright solitons in dipolar Bose-Einstein condensates with tilted dipoles

To Snake or Not to Snake in the Planar Swift–Hohenberg Equation

Localized radial solutions of the Swift–Hohenberg equation

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