Symmetric and antisymmetric nonlinear modes supported by dual local gain in lossy lattices

TL;DR

This paper investigates symmetric and antisymmetric nonlinear modes in a lossy lattice with dual local gain sites, revealing their analytical forms, stability properties, and dynamic transformations, relevant for optical waveguide applications.

Contribution

It introduces a new model with dual gain sites in lossy lattices, providing analytical solutions and stability analysis for symmetric and antisymmetric modes.

Findings

Stable and unstable mode subfamilies identified.

Symmetric modes can transform into stable antisymmetric modes.

Unstable modes lead to persistent breather states.

Abstract

We introduce a discrete lossy system, into which a double hot spot (HS) is inserted, i.e., two mutually symmetric sites carrying linear gain and cubic nonlinearity. The system can be implemented as an array of optical or plasmonic waveguides, with a pair of amplified nonlinear cores embedded into it. We focus on the case of the self-defocusing nonlinearity and cubic losses acting at the HSs. Symmetric localized modes pinned to the double HS are constructed in an implicit analytical form, which is done separately for the cases of odd and even numbers of intermediate sites between the HSs. In the former case, some stationary solutions feature a W-like shape, with a low peak at the central site, added to tall peaks at the positions of the embedded HSs. The special case of two adjacent HSs is considered too. Stability of the solution families against small perturbations is investigated in a numerical form, which reveals stable and unstable subfamilies. The instability of symmetric modes accounting for by an isolated positive eigenvalue leads to their spontaneous transformation into co-existing stable antisymmetric modes, while the instability represented by a pair of complex-conjugate eigenvalues gives rise to persistent breathers.