## 4 Discussion  
We have examined some properties of a coupled Schrodinger-elliptic system modeling optical waveguide arrays in a nematic liquid crystal substrate. The system is studied by first passing to an equivalent infinite system of discrete describing the interaction of Wannier mode amplitudes of the relevant physical quantities [1]. The Wannier basis functions are defined in terms of a periodic Schrodinger operator appearing in the system and must be computed numerically. The Wannier mode amplitudes are related to the observed quantities by quadrature formulas (21) that may be evaluated numerically or using approximations. The Wannier basis approach leads to the derivation of systems of discrete nonlinear Schrodinger (DNLS) equations by truncation to the mode amplitudes of the first bands of the periodic Schrodinger operator. We have shown that these systems are Hamiltonian. The proof uses the reality of Wannier functions, and we have also described an explicit construction of real Wannier functions. Finally we show that the linear part of the system of discrete equations is diagonalized by (trigonometric) plane waves. Bloch-Wannier analysis is a classical subject in theoretical physics, with well-known applications in classical and quantum mechanics [24; 25]. Recently it is increasingly applied to the study of nonlinear waves in inhomogeneous media [23; 26], and in the theory of homogenization [29; 30]. The paper considers a problem in nonlinear waves where the periodicity appears in the nonlinear interaction. The system includes a natural operator that allows the use of Bloch-Wannier analysis, but differs from the more commonly stud ied systems [7] where the linear part of the beam propagation equation is the periodic Schrodinger operator. In that case the band gap structure of the periodic Schrodinger operator allows us to justify the truncation to a model for the lowest band modes through a non-resonance argument [23]. In the present case the the most likely justification of DNLS models will involve truncation to a finite (possibly small) system of DNLS equations. This fact is also suggested by experimental results of [3]. Note that [3] consider a more complicated periodic operator in the director field equation that also includes the second (vertical) direction of the plane perpendicular to the optical axis. The present paper uses a simplification of the director field equation and takes advantage of the more developed Bloch analysis of the periodic Schrodinger operator. The idea is that the nonlinear effect of the transverse periodicity is already present in the simpler version. In addition, the more tractable Wannier-Bloch analysis of the simpler problem has allowed us to derive multi-band DNLS systems and to analyze their structure. This a first towards further analysis of such systems, e.g., along the lines of earlier studies of the Fratalocchi-Assanto model [8; 9]. While the paper considers a problem arising from nonlinear optics in liquid crystals, the combination of nonlocal nonlinearity and periodicity we consider may appear in other areas where similar systems are studied [14; 15]. The paper shows that the Wannier basis expansion is an effective tool for elucidating the structural features of simplified DNLS equations. We also saw that the Wannier-Bloch approach requires the numerical computation of several intermediate quantities. For instance, the computation of Wannier functions uses numerically computed Bloch functions and numerical integration over Bloch functions, while the nonlinearity of DNLS systems, as well as the linear interaction between bands also involves the evaluation of Wannier integrals. Possible simplifications may arise for some limits of \(V\)[1], and there are general ideas such a eliminating non-resonant interactions [23; 26] that can be examined further in this problem. but possibly less practical. Our view is that a good understanding of the structure of the Wannier coupled mode systems may allow us to analyze their dynamical properties without computing everything.