## 1 Introduction  
We study properties of an infinite system of discrete nonlinear Schrodinger (DNLS) equations that is equivalent to a coupled Schrodinger-elliptic system of partial differential equations with periodic coefficients. The system was derived in [1] as a model for the propagation of laser light in nematic liquid crystal substrates with a periodic structure in one of the directions normal to the optical axis. The model was originally motivated by experimental studies of such waveguide systems [2, 3, 4] and leads to extensions of a nonlocal DNLS equation of Fratalocchi and Assanto [5, 6]. The Fratalocchi-Assanto equation has a nonlocal nonlinearity that leads to new effects when compared to the cubic power DNLS model studied commonly in photonics and atomic physics [7]. These effects include non-monotonic amplitude profiles of static (breather) solutions, additional internal modes in the linearization around breathers [8, 9], and enhanced mobility of traveling localized solutions [10]. On the other hand, the mathematical justification of the Fratalocchi-Assanto model, in particular the question of how well it approximates the partial differential equations with periodic coefficients used to describe the underlying physics, is less studied. The present paper is a step in studying this problem. Schrodinger-elliptic systems of differential equations with a similar nonlocal structure in the nonlinear term arise in a variety of contexts. Examples from physics include Bose-Einstein condensates [11], thermal media [12], and matter-wave microwave systems [13]. The recent review [14] includes further examples describing laser beams in liquid crystals [5, 15, 16, 17]. A related area of application of such models concerns thermo-optical interactions induced by beams in liquid crystals [18, 19, 20]. The combination of nonlocal nonlinearity and spatial periodicity or more general inhomogeneity, and the analysis of relevant equations is therefore a problem of wider interest. The Schrodinger-elliptic differential equation we study describes the coupling of the laser field amplitude to the nematic crystal director angle. The derivation uses approximations of the coupled Oseen-Frank-Maxwell system for a linearly polarized beam [1]. The periodicity of the medium in the direction transverse to the propagation of the laser beam leads to an elliptic (Poisson-like) equation with periodic coefficients. Our approach is to expand the laser field and director angle in a Wannier function basis. The system is subsequently written as an infinite system of coupled DNLS equations for the Wannier mode amplitudes. The Wannier functions we use are defined in terms of a periodic Schrodinger operator appearing in the elliptic equation [1]. Note that the Wannier functions are integer translates of an infinite set of localized functions with an increasing degree of oscillation. Thus Wannier mode amplitudes give information on both the location and the spatial scale of images. Wannier and the related Bloch functions are a standard tool in the analysis of periodic Schrodinger operators [21; 22; 23; 24], and related linear problems in theoretical physics, e.g., solid state physics [25]. Wannier functions are increasingly used in the study of nonlinear waves in inhomogeneous media. The use of Wannier functions for deriving discrete Schrodinger equations for nonlinear wave systems with periodic coefficients was first proposed in [26] for the periodic Gross-Pitaevski equation (NLS with periodic potential). The Wannier expansion has been used to justify the approximation of this equation by the DNLS equation in the the tight binding approximation limit for the potential term in [23; 27]. Related systems where the theory applies are described in [11; 12]. In the present problem the Wannier basis leads to a heuristic derivation of the model of [6] and also allows us to derive more general DNLS-type equations and systems that include additional inter- and intra-band Wannier mode interactions [1].