Variational approximations to homoclinic snaking

Abstract

We investigate the snaking of localised patterns, seen in numerous physical applications, using a variational approximation. This method naturally introduces the exponentially small terms responsible for the snaking structure, that are not accessible via standard multiple-scales asymptotic techniques. We obtain the symmetric snaking solutions and the asymmetric 'ladder' states, and also predict the stability of the localised states. The resulting approximate formulas for the width of the snaking region show good agreement with numerical results.

arXiv comment4 pages, 3 figures, submitted

References in corpus (3)

• The emergence of a coherent structure for coherent structures: localized states in nonlinear systems
• Localized patterns and hole solutions in one-dimension extended sytem
• Snaking and isolas of localised states in bistable discrete lattices

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• Maxwell Fronts in the Discrete Nonlinear Schrödinger Equations with Competing Nonlinearities
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