Solitons in $\mathcal{PT}$-symmetric periodic systems with the quadratic nonlinearity

TL;DR

This paper explores soliton solutions in a one-dimensional $ ext{PT}$-symmetric system with quadratic nonlinearity, analyzing their stability and the effects of gain-loss balance on their properties.

Contribution

It introduces and investigates the stability of solitons, including embedded solitons, in a novel $ ext{PT}$-symmetric quadratic nonlinear system, highlighting the impact of gain-loss parameters.

Findings

Stable embedded solitons identified within the continuous spectrum.

The imaginary part of the potential significantly influences soliton stability.

Families of solitons bifurcate from spectral gap edges.

Abstract

We introduce a one-dimensional system combining the $\mathcal{PT}$-symmetric complex periodic potential and the $\chi ^{(2)}$ (second-harmonic-generating) nonlinearity. The imaginary part of the potential, which represents spatially separated and mutually balanced gain and loss, affects only the fundamental-frequency (FF) wave, while the real potential acts on the second-harmonic (SH) component too. Soliton modes are constructed, and their stability is investigated (by means of the linearization and direct simulations) in semi-infinite and finite gaps in the corresponding spectrum, starting from the bifurcation which generates the solitons from edges of the gaps' edges. Families of solitons embedded into the conttinuous spectrum of the SH component are found too, and it is demonstrated that a part of the families of these \textit{embedded solitons} (ESs) is stable. The analysis is focused on effects produced by the variation of the strength of the imaginary part of the potential, which is a specific characteristic of the $\mathcal{PT}$ system. The consideration is performed chiefly for the most relevant case of matched real potentials acting on the FF\ and SH components. The case of the real potential acting solely on the FF component is briefly considered too.