Localised states in an extended Swift-Hohenberg equation

TL;DR

This paper explores the effects of non-variational and non-conservative factors on localized states in an extended Swift-Hohenberg equation, analyzing pattern formation, bifurcations, and stability in a more complex setting than the traditional model.

Contribution

It introduces an extended Swift-Hohenberg model incorporating non-variational effects and provides a detailed analysis of pattern instability, bifurcation structures, and stability of localized states.

Findings

Normal form analysis of initial pattern instability

Bifurcation structure of large-amplitude localized states

Temporal stability of one-peak localized states

Abstract

Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure --- it is variational in time and conservative in space. In this paper we investigate an extended Swift-Hohenberg equation in which non-variational and non-conservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. Firstly we carry out the normal form analysis of the initial pattern forming instability that leads to small-amplitude localised states. Next we examine the bifurcation structure of the large-amplitude localised states. Finally we investigate the temporal stability of one-peak localised states. Throughout, we compare the localised states in the extended Swift-Hohenberg equation with the analogous solutions to the usual Swift-Hohenberg equation.