Pattern formation outside of equilibrium

Cross, M. C.; Hohenberg, P. C.

doi:10.1103/revmodphys.65.851

Cited by 7,323 publications

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References 966 publications

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“…where q = |q| and K denotes the Fourier transform of K. Note that σ(0) = 0 for all ρ 0 due to the conservative structure of (9) (Cross and Hohenberg, 1993). We determine the neutral stability curve q(ρ 0 ) by setting σ = 0 in (12).…”

Section: Conservation Of Momentsmentioning

confidence: 99%

A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions, as well, and is demonstrated via numerical simulations in two dimensions.

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Section: Conservation Of Momentsmentioning

confidence: 99%

A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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“…When the symmetries are purely spatial in nature (e.g. reflections, translations, rotations), these consequences have been extensively studied (see, for example, Golubitsky & Schaeffer 1985;Golubitsky, Stewart & Schaeffer 1988;Crawford & Knobloch 1991;Cross & Hohenberg 1993;Chossat & Iooss 1994;Iooss & Adelmeyer 1998;Chossat & Lauterbach 2000;Golubitsky & Stewart 2002). The system may also be invariant to the action of spatio-temporal symmetries.…”

Section: Introductionmentioning

confidence: 99%

Symmetry breaking of two-dimensional time-periodic wakes

2005

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A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-periodflip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

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“…To be specific, we focus on the Swift-Hohenberg equation, which has been proposed as a prototypical example for pattern forming systems, in areas as diverse as nonlinear optics [15], Rayleigh-Bénard convection [2], granular media [1], chemical reactions [14], liquid crystals and solidification; see [4] and references there. Most of those systems exhibit stationary stripe (or roll) patterns, that is, planar patterns which are independent of x and periodic in y, where x and y are coordinates in the plane of observation.…”

Section: Introductionmentioning

confidence: 99%

Interfaces between rolls in the Swift-Hohenberg equation

Hărăguş

Scheel²

2007

IJDSDE

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We study the existence of interfaces between stripe or roll solutions in the Swift-Hohenberg equation. We prove the existence of two different types of interfaces: corner-like interfaces, also referred to as knee solutions, and step-like interfaces. The analysis relies upon a spatial dynamics formulation of the existence problem and an equivariant center manifold reduction. In this setting, the interfaces are found as heteroclinic and homoclinic orbits of a reduced system of ODEs.

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Pattern formation outside of equilibrium

Cited by 7,323 publications

References 966 publications

A Nonlocal Continuum Model for Biological Aggregation

A Nonlocal Continuum Model for Biological Aggregation

Symmetry breaking of two-dimensional time-periodic wakes

Interfaces between rolls in the Swift-Hohenberg equation

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