A synthesis is presented of recent work by the authors and others on the formation of
localised patterns, isolated spots, or sharp fronts in models of natural processes governed …

For a Selkov--Schnakenberg model as a prototype reaction-diffusion system on two
dimensional domains, we use the continuation and bifurcation software\tt pde2path to …

Fully localised patterns involving cellular hexagons or squares have been found
experimentally and numerically in various continuum models. However, there is currently no …

In this paper, we present a methodology for establishing constructive proofs of existence of
smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation …

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 …

We study in this paper the existence of periodically modulated in one variable and localized
in another variable solutions to the cubic Swift–Hohenberg equation on the plane R 2. In the …

We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-
Hohenberg equation using a novel approach based on geometric blow-up. We show that …

We study homoclinic snaking in the cubic-quintic Swift–Hohenberg equation (SHE) close to
the onset of a subcritical pattern-forming instability. Application of the usual multiple-scales …

We present a rigorous numerical method for proving the existence of a localized radially
symmetric solution for a Ginzburg--Landau-type equation. This has a direct application to the …

An investigation is undertaken of coupled reaction–diffusion systems in one spatial
dimension that are able to support, in different regions of their parameter space, either an …