Variational approximations to homoclinic snaking in continuous and discrete systems

TL;DR

This paper develops a variational approach to analyze homoclinic snaking phenomena in continuous and discrete systems, capturing exponentially small pinning regions and aligning with existing asymptotic and numerical results.

Contribution

It introduces a variational method to approximate homoclinic snaking behavior, extending analysis to systems with small-scale patterns and discrete structures.

Findings

Results agree with exponential asymptotics for Swift-Hohenberg equation

Method accurately predicts pinning region width in discrete systems

Numerical simulations confirm the variational approximations

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.