Embedded solitons in second-harmonic-generating lattices

TL;DR

This paper investigates the existence, construction, and stability of embedded solitons in nonlinear discrete waveguide arrays with quadratic and cubic nonlinearities, revealing stable modes within the propagation band.

Contribution

It introduces analytical approximations and numerical methods to identify and analyze discrete embedded solitons in second-harmonic-generating lattices, including stability properties.

Findings

Discrete embedded solitons can be accurately approximated analytically.

Stable embedded solitons exist within the propagation band.

The DES branch extends into a semi-infinite gap as regular solitons.

Abstract

Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted, severally, for broad and narrow DESs, and in a systematic form with the help of numerical calculations. DESs of several types, including ones with identical and opposite signs of their fundamental-frequency and second-harmonic components, are produced. In the most relevant case of narrow DESs, the analytical approximation produces very accurate results, in comparison with the numerical findings. In this case, the DES branch extends from the propagation band into a semi-infinite gap as a family of regular discrete solitons. The study of stability of DESs demonstrates that, in addition to ones featuring the well-known property of semi-stability, there are linearly stable DESs which are genuinely robust modes.