PHYSICAL REVIEW E 91, 033207 (2015) Staggered parity-time-symmetric ladders with cubic nonlinearity Jennie D’Ambroise, 1 P. G. Kevrekidis, 2 and Boris A. Malomed 3 1 Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts 01002-5000, USA 2 Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305, USA 3 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel (Received 30 September 2014; published 25 March 2015) We introduce a ladder-shaped chain with each rung carrying a parity-time- (PT -) symmetric gain-loss dimer. The polarity of the dimers is staggered along the chain, meaning alternation of gain-loss and loss-gain rungs. This structure, which can be implemented as an optical waveguide array, is the simplest one which renders the system PT -symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schr ¨ odinger equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anticontinuum limit of uncoupled rungs and using the Newton’s method for continuation of the solutions with the increase of the inter-rung coupling, we construct families of PT -symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung and double ones are identified. Dynamics of unstable solitons is investigated too. DOI: 10.1103/PhysRevE.91.033207 PACS number(s): 05.45.Yv, 63.20.Ry I. INTRODUCTION A vast research area, often called discrete nonlinear optics, deals with evanescently coupled arrayed waveguides featuring material nonlinearity [1]. Discrete arrays of optical waveguides have drawn a great deal of interest not only because they intro- duce a vast phenomenology of the nonlinear light propagation, such as, e.g., the prediction [2] and experimental creation [3] of discrete vortex solitons, but also due to the fact that they offer a unique platform for emulating the transmission of electric signals in solid-state devices, which is obviously interesting for both fundamental studies and applications [1,4]. Furthermore, the flexibility of techniques used for the creation of virtual (photoinduced) [5] and permanently written [6] guiding arrays enables the exploration of effects which can be difficult to directly observe in other physical settings, such as Anderson localization [7]. Another field in which arrays of quasidiscrete waveguides find a natural application is the realization of the optical PT (parity-time) symmetry [8]. On the one hand, a pair of coupled nonlinear waveguides, which carry mutually balanced gain and loss, make it possible to realize PT -symmetric spatial or temporal solitons (if the waveguides are planar ones or fibers, respectively), which admit an exact analytical solution, including their stability analysis [9]. On the other hand, a PT -symmetric dimer, i.e., the balanced pair of gain and loss nodes, can be embedded, as a defect, into a regular guiding array, with the objective to study the scattering of incident waves on the dimer [10,11,13]. We note here in passing that sometimes, also the term “dipoles” may be used for describing such dimers; however, we will not make use of it here, to avoid an overlap in terminology with classical dipoles in electrodynamics as discussed, e.g., in [12]. Discrete solitons pinned to a nonlinear PT -symmetric defect have been reported too [13]. Such systems, although governed by discrete nonlinear Schr¨ odinger (DNLS) equations corresponding to non-Hermitian Hamiltonians, may generate real eigenvalue spectra (at the linear level), provided that the gain-loss strength does not exceed a critical value, above which the PT symmetry is broken [14] [self-defocusing nonlinearity with the local strength growing, in a one-dimensional (1D) system, from the center to periphery at any rate faster that the distance from the center, gives rise to stable fundamental and higher-order solitons with unbreakable PT -symmetry [15]]. One- and two-dimensional (1D and 2D) lattices, built of PT dimers, were introduced in Refs. [16,17] and [18], respectively. Discrete solitons, both quiescent and moving ones, were found in these systems [16,18]. In the continuum limit, those solitons go over into those in the above-mentioned PT -symmetric coupler [9]. Accordingly, a part of the soliton family is stable, and another part is unstable. Pairs of parallel and antiparallel coupled dimers, in the form of PT -symmetric plaquettes (which may be further used as building blocks for 2D chains), were investigated too [19,20]. The objective of this work is to introduce a staggered chain of PT -symmetric dimers, with the orientations of the dimers alternating between adjacent sites of the chain. This can also be thought of as an extension of a plaquette from Refs. [19,20] towards a lattice. While this ladder-structured lattice is not a full 2D one, it belongs to a class of chain systems which may be considered as 1.5D models [21]. As shown in Sec. II, where the model is introduced, the fundamental difference from the previously studied ones is the fact that such a system, although being nearly one dimensional, actually realizes the PT symmetry in the 2D form, with respect to both horizontal and vertical directions. In Sec. III, we start the analysis from the solvable anticontinuum limit (ACL) [22], in which the rungs of the ladder are uncoupled (in the opposite continuum limit, the ladder degenerates into a single NLS equation). Using parametric continuation from this limit makes it possible to construct families of discrete solitons in a numerical form. Such solution branches are initiated, in the ACL, by a single excited rung, as well as by the excitation confined to several rungs. The soliton stability is systematically analyzed in Sec. III too and, if the modes are identified as unstable, their evolution is examined to observe the instability development. The paper is concluded by Sec. IV, where also some directions for future study are presented. 1539-3755/2015/91(3)/033207(11) 033207-1 ©2015 American Physical Society