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Collisions of localized patterns in a nonvariational Swift-Hohenberg equation
Mathi Raja1, Adrian van Kan1, Benjamin Foster1
1Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA.
Physical Review. E
|July 19, 2023
Summary
This study investigates the cubic-quintic Swift-Hohenberg equation (SH35) with broken symmetry, revealing that localized structures (LSs) exhibit complex collision dynamics. These interactions lead to diverse bound states, predictable by a reduced ODE model, highlighting rich temporal dynamics.
Area of Science:
- Complex Systems and Nonlinear Dynamics
- Fluid Dynamics and Pattern Formation
- Computational Physics
Background:
- The cubic-quintic Swift-Hohenberg equation (SH35) models convective systems with midplane reflection symmetry, such as binary fluid convection.
- Breaking this symmetry introduces nonvariational terms, leading to asymmetric spatially localized structures (LSs).
Purpose of the Study:
- To investigate the dynamics of asymmetric localized structures (LSs) in a nonvariational cubic-quintic Swift-Hohenberg equation (SH35).
- To analyze collision scenarios between these structures and understand the resulting bound states.
- To develop and validate a reduced ordinary differential equation (ODE) model for LS interactions.
Main Methods:
- Numerical continuation and extensive direct numerical simulations (DNSs) were employed.
- Asymptotic analysis was used to predict drift velocities of LSs.
- A reduced ODE model was formulated and validated against DNS data using gradient descent optimization.
Main Results:
- Nonvariational SH35 supports asymmetric LSs with predictable drift velocities.
- Collisions between LSs are inelastic, forming bound states that can be longer or shorter than the initial structures.
- A reduced ODE model accurately captures the dynamics for simple bound states but shows limitations with wavelength changes.
Conclusions:
- The stability of bound states, not the Maxwell point, governs collision outcomes in nonvariational SH35.
- Complex parameter-space structures (isolas) describe multipulse bound states.
- The reduced ODE model provides a significant quantitative description of LS interactions, revealing net attractive or repulsive behaviors.
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