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Variational approximations to homoclinic snaking in continuous and discrete systems
1School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2012
Summary
Localized structures arise from pinning mechanisms, often with exponentially small pinning regions. A variational method successfully captures this behavior in both continuous and discrete nonlinear systems, aligning with prior findings and simulations.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
- Pattern Formation
Background:
- Localized structures are observed in diverse systems, often linked to pinning mechanisms.
- A key challenge is analyzing the exponentially small width of pinning regions when length scales separate.
- Standard asymptotic methods struggle to address this scale separation.
Purpose of the Study:
- To develop and apply a variational method for analyzing localized structures with exponentially small pinning regions.
- To validate the method against established techniques and numerical simulations.
- To demonstrate the broad applicability of the variational approach.
Main Methods:
- A variational method is employed to analyze the behavior of localized structures.
- The method is applied to the quadratic-cubic Swift-Hohenberg equation.
- The approach is also tested on a discrete system with cubic-quintic nonlinearity.
Main Results:
- The variational method successfully reproduces the exponentially small width of pinning regions.
- Results for the Swift-Hohenberg equation align with recent exponential asymptotics studies.
- The method shows good agreement with numerical simulations for the discrete nonlinear system.
Conclusions:
- The variational method provides an effective tool for studying localized structures with scale separation.
- This approach offers an alternative to complex asymptotic techniques.
- The findings highlight the robustness of the variational method across different system types.
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