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Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system.

D Avitabile1, M Desroches2,3, E Knobloch4

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This study analyzes pattern formation in a nonlinear system, identifying two types of stationary fronts. Their behavior depends on boundary conditions and system instabilities, offering insights into complex spatial structures.

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Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Mathematical physics

Background:

  • Subcritical pattern-forming systems exhibit complex behaviors.
  • Nonlinear advection in bounded domains presents analytical challenges.
  • Understanding stationary fronts is crucial for predicting system dynamics.

Purpose of the Study:

  • To analyze a subcritical pattern-forming system with nonlinear advection in a bounded domain.
  • To identify and characterize different types of stationary fronts.
  • To elucidate the role of boundary conditions and instabilities in front dynamics.

Main Methods:

  • Recasting the system as a slow-fast system in space.
  • Applying geometric singular perturbation theory.
  • Utilizing numerical continuation and Monte Carlo simulations.
  • Analyzing partial differential equations.

Main Results:

  • Two types of stationary fronts were identified, linked to convective and absolute instabilities.
  • One front type depends on upstream boundary conditions, explaining noise-sustained structures.
  • The other front type corresponds to a global mode, present even with zero upstream conditions.
  • Canard trajectories' role in transitions and stability of spatial structures were clarified.

Conclusions:

  • Front location in the convective regime is sensitive to upstream boundary conditions.
  • Statistical properties under stochastic boundary conditions can be interpreted via the deterministic slow-fast system.
  • The study provides a quantitative understanding of noise-sustained structures and spatial dynamics.