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Published on: December 4, 2017
Localized states in the generalized Swift-Hohenberg equation
1Department of Physics, University of California, Berkeley, 94720, USA. burkej8@socrates.berkeley.edu
Summary
The Swift-Hohenberg equation reveals numerous stable localized states due to homoclinic snaking. Numerical methods show how these states change and lose stability over time.
Area of Science:
- Nonlinear dynamics
- Mathematical physics
Background:
- The Swift-Hohenberg equation models pattern formation.
- It is known to possess spatially localized solutions.
Purpose of the Study:
- To investigate the stable spatially localized states in the Swift-Hohenberg equation with quadratic and cubic nonlinearities.
- To understand the phenomenon of homoclinic snaking and its role in forming these states.
Main Methods:
- Numerical computations were employed to analyze the localized solutions.
- Direct simulations in time were used to observe the evolution of states past their stability threshold.
Main Results:
- A rich variety of stable, spatially localized states were identified.
- The study illustrated changes in localized solutions with increasing spatial extent.
- Stability properties of these states were determined.
Conclusions:
- Homoclinic snaking is a key mechanism for the existence of these localized states.
- The dynamics of localized states, including their loss of stability, were characterized.
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