Localized radial roll patterns in higher space dimensions

TL;DR

This paper investigates the structure of localized radial roll patterns in higher dimensions, revealing complex bifurcation behaviors such as isolas and snaking through perturbation analysis.

Contribution

It provides a perturbation analysis of localized radial roll solutions in dimensions slightly above one, elucidating features observed in two-dimensional cases.

Findings

Branches exhibit both isolas and snaking in higher dimensions.

Perturbation analysis explains complex bifurcation structures.

Insights connect one-dimensional and planar localized patterns.

Abstract

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves ("isolas"), or the length increases to infinity so that branches are unbounded in function space ("snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyse the structure of branches of localized radial roll solutions in dimension 1+$\varepsilon$, with $0<\varepsilon\ll1$, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.