From Snaking to Isolas: A One-Active-Site Approximation in Discrete Optical Cavities

TL;DR

This paper studies discrete optical cavities with Kerr nonlinearity, revealing how localized solitons form and transition from snaking bifurcations to isolas, and introduces a one-active-site approximation that accurately captures these phenomena.

Contribution

It introduces a novel one-active-site approximation for analyzing localized solutions in discrete optical cavities, especially near the anti-continuum limit.

Findings

The approximation accurately predicts soliton stability.

Snaking bifurcations transition into isolas near the anti-continuum limit.

Numerical results agree well with the analytical approximation.

Abstract

We investigate time-independent solutions of a discrete optical cavity model featuring saturable Kerr nonlinearity, a discrete version of the Lugiato-Lefever equation. This model supports continuous wave (uniform) and localized (discrete soliton) solutions. Stationary bright solitons arise through the interaction of dark and bright uniform states, forming a homoclinic snaking bifurcation diagram within the Pomeau pinning region. As the system approaches the anti-continuum limit (weak coupling), this snaking bifurcation widens and transitions into $\subset$-shaped isolas. We propose a one-active-site approximation that effectively captures the system's behavior in this regime. The approximation also provides insight into the stability properties of soliton states. Numerical continuation and spectral analysis confirm the accuracy of this semianalytical method, showing excellent agreement with the full model.