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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.
Physical Review. E
|September 11, 2019
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.
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.
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