## 1 An Overview of the Topic and Its Ramifications
### Linear Waveguides
A natural generalization of single-channel waveguides is provided by a _coupler_, which may be considered as a set of two parallel waveguiding cores, coupled in the transverse direction by tunneling of guided wave fields steered by each tunnel in the longitudinal direction. The respective system of coupled equations for amplitudes \(u\) and \(v\) of electromagnetic waves in the two cores is [25] (cf. Equation (2)):  
\[ \begin{array}{rcl}i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{ \partial^{2}u}{\partial x^{2}}+\kappa v-U(x)u&=&0.\\ \\ i\frac{\partial v}{\partial z}+\frac{1}{2}\frac{\partial^{2}v}{ \partial x^{2}}+\kappa u-U(x)v&=&0,\end{array}  \tag{4}\]  
where \(\kappa\) is the coefficient of the linear inter-core coupling. The next step is to consider _arrayed systems_, composed of many parallel guiding cores, which are also coupled in the transverse direction(s) by the tunneling of longitudinally guided wave fields (planar and bulk arrays have, respectively, one or two transverse coordinates). The simplest model of such a guiding medium is provided by the two- or three-dimensional scaled Schrodinger equation with a periodic transverse potential, which represents the (idealized) structure of the multi-core bundle:  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\left(\frac{\partial^{2}u}{ \partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right)-\epsilon\left[ \cos\left(\frac{2\pi x}{l}\right)+\cos\left(\frac{2\pi y}{l}\right)\right]u=0. \tag{5}\]  
Here \(l\) is the array's period (defined in scaled units, in which Equation (5) is written), and \(2\epsilon\) is the scaled depth of the effective trapping potential. In particular, in optics bulk arrays have been created as permanent structures by burning (also by means of an optical technology) a large number of parallel guiding cores in a bulk piece of silica [26]. As concerns planar guiding arrays, an interesting ramification of the topic is the propagation of optical waves in such arrays made with a curved shape [27]. On the other hand, a technology for the creation of reconfigurable _virtual_ conduit patterns in the form of photonic lattices was elaborated for photorefractive materials [28]. The latter technology makes use of the fundamental property of the photorefractive materials, in which the propagation conditions for light with ordinary and extraordinary polarizations are linear and nonlinear, respectively. To create a photonic lattice, the experimentalist first illuminates the sample by counterpropagating pairs of mutually coherent laser beams in the ordinary polarization, which create a classical interference pattern in the photorefractive crystal, which is an effectively linear medium for these beams. Next, a probe beam is launched, with the extraordinary polarization in the transverse direction. Due to its inherent nonlinearity, the probe beam is affected by the originally created photonic lattice, as if it is a material structure that creates a spatially periodic modulation of the local refractive index in the transverse directions; i.e., essentially, another version of the multi-core guiding structure. The propagation of light or waves of a different physical nature in arrays with weak coupling between guiding cores may be naturally approximated by the discrete Schrodinger equation. The basic realization of such a medium is represented by planar arrays of parallel optical waveguides coupled by evanescent waves penetrating dielectric barriers separating individual cores, the basic model being a scaled discrete version of Equation (2):  
\[i\frac{du_{n}}{dz}+\frac{1}{2}\left(u_{n+1}+u_{n-1}-2u_{n}\right)-U_{n}u_{n}=0, \tag{6}\]  
where the discrete coordinate, \(n\), which replaces \(x\), is the number of the guiding core in the array. The study of light propagation in various multi-core systems--which may be approximated by lattice models similar to Equation (6)--is a vast area known as _discrete optics_[29].