Mathematics & Statistics Departmental Research Seminar

Spatial Localization in Discrete and Continuous Media

Abstract:

Spatially localized patterns are solutions, often of differential equations, where a spatial pattern is embedded in a quiescent state. These types of structures occur in many different applications including liquid crystals, sunspots, cell patterning, vegetation patches, and much more. These localized patterns have been well-studied in many contexts, including existence and bifurcation structure in discrete and continuous settings, with some extensions to higher dimensions and stability.

In this talk I will focus on two settings:

1. Stability of localized patterns in lattice dynamical systems: I will present a general theoretical framework to analyze the spectral stability of localized steady states in one-dimensional and multi-dimensional rectangular lattices. The approach leverages the properties of front and back solutions, combined with a discrete Evans function formulation. This framework applies to solutions in single and multiple dimensions, including oscillatory and multi-pulse configurations.

2. Localized patterns in 3D. I will finish by discussing some first results on the bifurcation structure of localized patterns in continuous 3D media.