Staggered $\mathcal{PT}$-symmetric ladders with cubic nonlinearity

Abstract

We introduce a ladder-shaped chain with each rung carrying a $\mathcal{PT}$ -symmetric gain-loss dipole. The polarity of the dipoles is staggered along the chain, meaning that a rung bearing gain-loss is followed by one bearing loss-gain. This renders the system $\mathcal{PT}$-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schr\"{o}dinger (DNLS) equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anti-continuum limit of uncoupled rungs and using the Newton's method for identifying solutions and parametric continuation in the inter-rung coupling for following the associated branches, we construct families of $\mathcal{PT}$-symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung, as well as ones from multiple rungs are identified. Dynamics of unstable solitons is presented too.