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arXiv · 1105.5268

Variational approximations to homoclinic snaking in continuous and discrete systems

P. C. Matthews·H. Susanto
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Abstract

Localised structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of lengthscales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behaviour can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. Secondly, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.

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P. C. Matthews, H. Susanto. 2011-05-26. Variational approximations to homoclinic snaking in continuous and discrete systems. https://doi.org/10.1103/physreve.84.066207

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