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Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation.

Edgar Knobloch1, Hannes Uecker2, Daniel Wetzel2

  • 1Department of Physics, University of California, Berkeley, California 94720, USA.

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Summary

This study numerically investigates the cubic-quintic-septic Swift-Hohenberg (SH357) equation. It reveals rich pattern formation, including bistability between stripe amplitudes and complex localized structures.

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Area of Science:

  • Nonlinear Dynamics
  • Pattern Formation
  • Computational Physics

Background:

  • The Swift-Hohenberg equation is a fundamental model for pattern formation in spatially extended systems.
  • Investigating complex variants like the cubic-quintic-septic (SH357) equation is crucial for understanding diverse pattern behaviors.
  • Bounded one-dimensional domains present unique challenges and opportunities for pattern selection.

Purpose of the Study:

  • To numerically explore pattern formation in the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded 1D domains.
  • To identify and characterize different types of stationary and dynamic patterns, including stripe phases and localized structures.
  • To analyze the bifurcations and stability of these patterns, particularly focusing on bistability phenomena.

Main Methods:

  • Numerical simulations were employed to solve the SH357 equation.
  • Bifurcation analysis using numerical continuation techniques was performed.
  • The role of conserved quantities, such as the spatial Hamiltonian, was investigated to understand observed phenomena.

Main Results:

  • Supercritical bifurcation of stripes with wave number k≈1 from the zero state, forming S-shaped branches and leading to bistability between small and large amplitude stripes.
  • Observation of stationary heteroclinic connections (fronts) between these stripe states within the bistability range.
  • Identification of localized defect-like structures that exhibit snaking behavior or reside on isolas, alongside connections to homogeneous states and stable multi-patch patterns.

Conclusions:

  • The SH357 equation exhibits remarkable richness in pattern formation on bounded 1D domains.
  • Bistability, complex localized structures, and diverse stable steady states are key features of this system.
  • Numerical continuation and conserved quantities provide valuable insights into the observed complex dynamics and bifurcations.