Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear $\mathcal{PT}$-symmetric defect

TL;DR

This paper constructs and analyzes discrete solitons in a waveguide array with a nonlinear $ ext{PT}$-symmetric defect, exploring their existence, stability, and scattering properties.

Contribution

It provides exact analytical solutions for discrete solitons in a $ ext{PT}$-symmetric nonlinear defect and investigates their stability and scattering behavior.

Findings

Fundamental solitons exist above a power threshold in tightly knit lattices.

Dipole modes exist above a finite power threshold.

Stability regions depend on total power and coupling strength.

Abstract

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded $\mathcal{PT}$-symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.