Broken and unbroken PT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mathcal {PT}}$$\end{document}-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

In this letter, we obtain multi-soliton solutions in terms of ratio of ordinary determinants for semi-discrete nonlocal nonlinear Schrödinger (sd-NNLS) equation by employing the Darboux transformation. We construct explicit expressions of single and double soliton solutions in zero background. We obtain symmetry-broken and symmetry-unbroken soliton solutions of sd-NNLS equation by using appropriate eigenfunctions. We notice that for symmetry non-preserving case, the potential term exhibits stable structure whereas individual fields display instability. We also obtain blowup or oscillating singular-type soliton solutions for symmetry-preserving case.