Formation of localized structures in bistable systems through nonlocal spatial coupling I: General framework

TL;DR

This paper presents a general framework for understanding how nonlocal spatial coupling influences the formation of localized structures in bistable systems, highlighting the role of spatial oscillations and specific transition points.

Contribution

It introduces a novel theoretical framework that explains how nonlocal interactions induce localized structures via front oscillations and pinning mechanisms in extended systems.

Findings

Nonlocal terms can induce spatial oscillations in front tails.

Localized structures emerge from pinning between oscillatory fronts.

Transitions between different front behaviors are organized by codimension points.

Abstract

The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in 1-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that non local terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, emerging from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 point s and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.