Variational approximations to homoclinic snaking

TL;DR

This paper uses a variational approximation to analyze homoclinic snaking in localized patterns, capturing exponentially small effects and predicting stability, with results aligning well with numerical data.

Contribution

It introduces a variational method that naturally accounts for exponentially small terms in homoclinic snaking, providing new analytical insights and stability predictions.

Findings

Derived formulas for the width of the snaking region

Predicted stability of localized states

Achieved good agreement with numerical results

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.