Homoclinic snaking in the discrete Swift-Hohenberg equation

TL;DR

This paper studies how discretizing the Swift-Hohenberg equation affects its bifurcation behavior, revealing different solution patterns and stability properties across coupling regimes using numerical and theoretical methods.

Contribution

It provides a detailed analysis of the discretized Swift-Hohenberg equation's bifurcation structure, including the effects of coupling strength on snaking and solution stability.

Findings

Multiple Maxwell points can occur in the intermediate coupling region.

Discretization leads to irregular snaking and isolas in certain regimes.

Theoretical analysis confirms stability patterns in the weak coupling region.

Abstract

We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localised and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.