Temporally Harmonic Oscillons in Newtonian Fluids

Arbell, H.; Fineberg, Jay

doi:10.1103/physrevlett.85.756

Cited by 104 publications

(112 citation statements)

References 22 publications

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“…Recently, in a nematic liquid crystal light valve with optical feedback it has been found experimentally a different type of localized states, appearing as a large amplitude peaks nucleating over a lower amplitude pattern and therefore called localized peaks [12]. Similar observations have been reported in a Newtonian fluid when non linear surface waves are parametrically excited with two frequencies [13] and in numerical simulations of an atomic vapor with optical feedback [14]. Recently, longitudinal modes with localized peaks over a spatially modulated background have been shown in numerical simulations of Maxwell-Bloch equations for a semiconductor laser [15].…”

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confidence: 63%

Localized States in Bistable Pattern-Forming Systems

Bortolozzo

Clerc²,

Falcón³

et al. 2006

Phys. Rev. Lett.

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We present an unifying description close to a spatial bifurcation of localized states, appearing as large amplitude peaks nucleating over a pattern of lower amplitude. Localized states are pinned over a lattice spontaneously generated by the system itself. We show that the phenomenon is generic and requires only the coexistence of two spatially periodic states. At the onset of the spatial bifurcation, a forced amplitude equation is derived for the critical modes, which accounts for the appearance of localized peaks. In one-dimensional systems, localized patterns can be described as homoclinic orbits passing close to a spatially oscillatory state and converging to an homogeneous state [9,10], whereas domains are seen as heteroclinic trajectories joining the fixed points of the corresponding dynamical system [11]. Recently, in a nematic liquid crystal light valve with optical feedback it has been found experimentally a different type of localized states, appearing as a large amplitude peaks nucleating over a lower amplitude pattern and therefore called localized peaks [12]. Similar observations have been reported in a Newtonian fluid when non linear surface waves are parametrically excited with two frequencies [13] and in numerical simulations of an atomic vapor with optical feedback [14]. Recently, longitudinal modes with localized peaks over a spatially modulated background have been shown in numerical simulations of Maxwell-Bloch equations for a semiconductor laser [15].All these different types of localized states appear over a patterned background and thus constitute a different class of structures with respect to the ones appearing over an uniform background. The aim of this manuscript is to show that localized peaks are a generic class of localized states, appearing whenever a pattern forming system exhibits coexistence of two spatially periodic states. The mechanisms that originate this circumstances are more than a few, for instance, one can consider a multi-stable system, which shows two consecutive spatial bifurcations to different states when one parameter is changed. There is a large number of physical systems that display this kind of behavior, therefore there is a vast number of possible models. In order to derive an unifying and simple description of localized peaks, we develop a theoretical model for one-dimensional spatially extended systems close to a spatial bifurcation. The model, which shows coexistence between different patterns and stable front solutions between them, is based on an amplitude equation that includes a spatial parametric forcing. This extension with respect to conventional amplitude equations, allows to describe localized patterns and to account for the main properties of these solutions. The model includes the interaction of the slowly varying envelope with the small scale of the underlying pattern solution [16], well-known as the non-adiabatic effect [17,18].

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confidence: 63%

Localized States in Bistable Pattern-Forming Systems

Bortolozzo

Clerc²,

Falcón³

et al. 2006

Phys. Rev. Lett.

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“…Such solutions have been observed in many different experiments, often under subharmonic forcing, and we refer the reader to [2,22,38,40] for examples. In one space dimension, bifurcations to oscillons have been studied recently in [44] near spatially homogeneous Hopf bifurcations with subharmonic forcing.…”

Section: Summary and Open Problemsmentioning

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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“…Time-periodic localised states called oscillons were first observed in parametrically driven granular media [44] and subsequently in parametrically driven liquids [1,29]. In many of these systems, the structures form bound states due to mutual attraction, often with an underlying hexagonal structure [1,2,7]; when parameters are varied, these structure may dissociate or melt [7].…”

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.

104

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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Temporally Harmonic Oscillons in Newtonian Fluids

Cited by 104 publications

References 22 publications

Localized States in Bistable Pattern-Forming Systems

Localized States in Bistable Pattern-Forming Systems

Localized radial solutions of the Swift–Hohenberg equation

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

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