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Area of Science:

  • Complex systems dynamics
  • Nonlinear physics
  • Pattern formation

Background:

  • Pattern formation in physical, biological, and chemical systems arises from instabilities.
  • Simultaneous instabilities lead to complex patterns beyond simple superposition.
  • Understanding coupled instabilities is crucial for predicting emergent behaviors.

Purpose of the Study:

  • Investigate pattern formation from coupled, asymmetric instabilities.
  • Analyze dynamics in non-conserved (Swift-Hohenberg) and conserved (Cahn-Hilliard) systems.
  • Characterize emergent spatio-temporal patterns and phase diagrams.

Main Methods:

  • Developed two models with asymmetrically coupled Swift-Hohenberg or Cahn-Hilliard equations.
  • Employed linear stability analysis to derive phase diagrams.
  • Conducted numerical simulations to explore nonlinear dynamics.

Main Results:

  • Swift-Hohenberg models exhibit Turing-wave instability coexistence and stationary/traveling wave regions.
  • Numerical simulations reveal complex patterns: traveling waves with Turing domains, chaos, and hysteresis.
  • Cahn-Hilliard models show arrested coarsening and emergence of periodic patterns with logarithmic divergence of domain length.

Conclusions:

  • Coupled instabilities generate rich spatio-temporal dynamics and complex patterns.
  • The interplay between different instabilities leads to phenomena not predictable from individual behaviors.
  • Weak coupling in conserved systems can significantly alter pattern evolution and arrest coarsening.