Dan J. Hill

Universität des Saarlandes | UKS · Department of Mathematics

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in var...

Localized radial patterns have been observed in the vegetation of semi-arid ecosystems, often as localized patches of vegetation or in the form of ‘fairy circles’. We consider stationary localized radial solutions to a reduced model for dryland vegetation on flat terrain. By considering certain prototypical pattern-forming systems, we prove the exi...

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven...

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern’s centre (the core region) and decay exponentially away from the pattern’s cen...

Localised radial patterns have been observed in the vegetation of semi-arid ecosystems, often as localised patches of vegetation or in the form of ‘fairy circles’. We consider stationary localised radial solutions to a reduced model for dryland vegetation on flat terrain. By considering certain prototypical pattern-forming systems, we prove the exi...

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's cen...