The emergence of a coherent structure for coherent structures: localized states in nonlinear systems

TL;DR

This paper reviews recent progress in understanding localized states in nonlinear systems, highlighting their formation, applications, and future research directions in the study of coherent structures.

Contribution

It synthesizes recent advances in localized states within pattern-forming nonlinear systems and proposes new application areas and future research directions.

Findings

Localized states form patches of regular pattern in homogeneous backgrounds

Recent progress has clarified the formation mechanisms of localized states

Potential applications span multiple scientific disciplines

Abstract

Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific sub-class of such problems, where a pattern-forming, or `Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states, proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally.