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Localized states in bistable pattern-forming systems.

U Bortolozzo1, M G Clerc, C Falcon

  • 1Institut Non Linéaire de Nice, UMR 6618, 1361 Route des Lucioles, 06560 Valbonne, France.

Physical Review Letters
|June 29, 2006
PubMed
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Localized states emerge from spatial bifurcations, forming large amplitude peaks on a lower amplitude pattern. This generic phenomenon, pinned by a self-generated lattice, requires coexisting periodic states.

Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Mathematical physics

Background:

  • Localized states are crucial in various physical systems.
  • Understanding their nucleation and stability is a key challenge.
  • Previous models often lack a unifying description for generic cases.

Purpose of the Study:

  • To provide a unifying description of localized states near spatial bifurcations.
  • To explain the nucleation of large amplitude peaks over lower amplitude patterns.
  • To demonstrate the generic nature of this phenomenon.

Main Methods:

  • Analysis of spatial bifurcation.
  • Derivation of a forced amplitude equation for critical modes.
  • Investigation of systems with coexisting spatially periodic states.

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Main Results:

  • Localized states are observed to nucleate as large amplitude peaks.
  • These states are pinned by a spontaneously generated lattice.
  • The phenomenon is shown to be generic, requiring only two coexisting periodic states.
  • A forced amplitude equation accurately describes the onset of localized peaks.

Conclusions:

  • A unified framework for understanding localized states near spatial bifurcations is established.
  • The spontaneous generation of pinning lattices is a key feature.
  • The derived amplitude equation provides predictive power for localized peak formation.