Localized Patterns

TL;DR

This paper reviews recent mathematical and numerical advances in understanding localized patterns, which are coherent structures in various media, highlighting progress, challenges, and open problems especially in higher dimensions.

Contribution

It provides a comprehensive overview of recent developments in the analysis of localized patterns, emphasizing mathematical tools and the increasing complexity with higher spatial dimensions.

Findings

Mathematical understanding decreases with increasing spatial dimension.

Numerical techniques have significantly advanced the study of localized patterns.

Open problems remain in higher-dimensional pattern analysis.

Abstract

Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.