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Swift-Hohenberg equation with broken reflection symmetry
J Burke1, S M Houghton, E Knobloch
1Center for BioDynamics, Boston University, Boston, Massachusetts 02215, USA. jb@math.bu.edu
Breaking spatial reversibility in the bistable Swift-Hohenberg equation causes localized states to drift. This disrupts the snakes-and-ladders structure, forming isolas and altering system dynamics.
Area of Science:
- Nonlinear dynamics
- Mathematical physics
- Pattern formation
Background:
- The bistable Swift-Hohenberg equation exhibits spatially localized solutions forming a snakes-and-ladders structure.
- This structure arises from homoclinic snaking, dependent on the equation's spatial reversibility.
Purpose of the Study:
- To investigate the impact of breaking spatial reversibility on the snakes-and-ladders structure.
- To analyze the resulting changes in localized states and system dynamics.
Main Methods:
- Numerical simulations to observe system behavior.
- Analytical techniques to understand the underlying dynamics.
- Exploration of varying degrees of reversibility breaking.
Main Results:
- Localized states were found to drift when spatial reversibility is broken.
- The snakes-and-ladders structure disintegrates into a stack of isolas.
- The evolution of this new structure with increasing reversibility breaking was mapped.
Conclusions:
- Breaking spatial reversibility fundamentally alters the organization of localized states.
- The transition from snakes-and-ladders to isolas signifies a significant change in system behavior.
- Understanding these dynamics is crucial for systems with broken spatial symmetry.
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