Computing has revolutionised the study of complex nonlinear systems, both by allowing us
to solve previously intractable models and through the ability to visualise solutions in …

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 …

Isolated patches of spatially oscillating pattern have been found to emerge near a pattern-
forming instability in a wide variety of experiments and mathematical models. However …

The emergence of localized radial patterns from a Turing instability has been well studied in
two-and three-dimensional settings and predicted for higher spatial dimensions. We prove …

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 …

Localised patterns are often observed in models for dryland vegetation, both as peaks of
vegetation in a desert state and as gaps within a vegetated state, known as 'fairy circles' …

Spatially localized patterns offer fascinating insights into the emergence of organized
behaviors, from weather fronts and desert vegetation to stop-and-go traffic waves and the …

In this article, we develop a method to perform constructive proofs of existence of smooth
planar localized patterns in the Gray-Scott model. Specifically, we construct a natural Hilbert …

Méthodes spectrales et preuves assistées par ordinateur pour l'étude d'EDP semi-linéaires
sur Rm Page 1 Méthodes spectrales et preuves assistées par ordinateur pour l’étude d’EDP …