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Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework.

Pere Colet1, Manuel A Matías1, Lendert Gelens2

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Nonlocal spatial coupling can create localized structures in one-dimensional systems by inducing oscillations in front tails. This phenomenon emerges from pinning effects between two fronts, offering new possibilities for structure formation.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Condensed Matter Physics

Background:

  • Extended systems often exhibit complex behaviors like localized structures.
  • Understanding the role of spatial coupling is crucial for predicting system dynamics.
  • Nonlocal interactions introduce long-range influences that can significantly alter system behavior.

Purpose of the Study:

  • To investigate the influence of nonlocal spatial coupling on localized structures in 1D systems.
  • To develop a general framework for understanding nonlocality's effect on front dynamics.
  • To identify the conditions and transitions leading to the formation of localized structures.

Main Methods:

  • Analysis of a real field model with linear nonlocal coupling.
  • Application of spatial dynamics theory.
  • Investigation of front dynamics and stability analysis.
  • Identification of phase transitions in parameter space.

Main Results:

  • Nonlocal coupling can induce spatial oscillations in the tails of fronts connecting stable states.
  • Localized structures emerge from the pinning of two such oscillatory fronts.
  • Three critical transitions define the oscillatory front region: modulational instability, Belyakov-Devaney transition, and a finite-wavelength oscillation crossover.
  • These transitions are organized by codimension 2 and 3 points.

Conclusions:

  • Nonlocal spatial coupling is a key mechanism for generating localized structures in 1D systems.
  • The study provides a comprehensive understanding of the transitions governing oscillatory fronts and structure formation.
  • Tuning nonlocal coupling parameters allows for controlled creation of localized structures.