## 1 An Overview of the Topic and Its Ramifications
### Nonlinear Waveguides  
In many situations, tightly confined guided waves propagating in conduits with a small effective cross-sectional area acquire high amplitudes, which is a source of a great many fascinating nonlinear effects. In particular, waveguides often provide a combination of the nonlinearity, group-velocity dispersion, and low (or sometimes completely negligible) losses which are necessary ingredients for the creation of _solitons_ (robust self-trapped solitary waves). The simplest and ubiquitous model of the nonlinear wave propagation is based on the nonlinear Schrodinger equation (NLSE), which in the simplest case includes a cubic term. In optics, this term represents the Kerr effect; i.e., nonlinear self-focusing (or, sometimes, self-defocusing) of light in the dielectric medium. The accordingly amended linear Schrodinger Equation (2) becomes the NLSE:  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{\partial x^{2} }-U(x)u+\sigma|u|^{2}u=0, \tag{7}\]  
where \(\sigma=+1\) and \(-1\) corresponds, respectively, to the self-focusing and defocusing nonlinearity; i.e., self-attraction and self-repulsion of light in the nonlinear medium. Equations (4) and (5) each acquire the same cubic terms as in Equation (7). In particular, the nonlinear version of Equation (4),  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{ \partial x^{2}}+\kappa v-U(x)u+\sigma|u|^{2}u = 0, \tag{8}\] \[i\frac{\partial v}{\partial z}+\frac{1}{2}\frac{\partial^{2}v}{ \partial x^{2}}+\kappa u-U(x)v+\sigma|v|^{2}v = 0,\]  
is the basic model of nonlinear couplers, their remarkable property being _spontaneous symmetry breaking_ in the case of self-focusing in the parallel-coupled cores, \(\sigma=+1\)[25; 30; 31]. A remarkable property of the one-dimensional NLSE in the absence of the potential (\(U=0\) in Equation (7)) is that it is an _integrable equation_ for which a very large number of exact solutions--including multi-soliton states--can be produced by means of a mathematical technique based on the _inverse scattering transform_[32; 33; 34]. These are bright and dark solitons in the cases of self-focusing and defocusing, respectively. In particular, the exact bright-soliton solution to Equation (7) with \(\sigma=+1\) and \(U=0\) is  
\[u(x,z)=\eta\exp\left(\frac{i}{2}\left(\eta^{2}-c^{2}\right)z+icx\right)\ \text{ sech}\left(\eta(x-cz)\right., \tag{9}\]  
where \(\eta\) and \(c\) are, respectively, the arbitrary amplitude and velocity of the soliton (in fact, in the spatial domain--in terms of which Equation (7) is written--the soliton represents a self-trapped light beam, and accordingly \(c\) is not a velocity, but rather a parameter which determines the tilt of the beam in the \((x,z)\) plane). The discrete Schrodinger Equation (6) also has its natural nonlinear counterpart in the form of discrete NLSE:  
\[i\frac{du_{n}}{dz}+\frac{1}{2}\left(u_{n+1}+u_{n-1}-2u_{n}\right)-U_{n}u_{n}+ \sigma|u_{n}|^{2}u_{n}=0; \tag{10}\]  
i.e., a discrete version of NLSE (7). The discrete NLSE gives rise to discrete solitons and their bound states, which cannot be found in an exact form, but may be efficiently produced by numerical and approximate analytical methods [35]. The propagation of nonlinear waves in discrete waveguiding arrays was the subject of numerous theoretical and experimental works [29; 36]. The multidimensional extension of the NLSE also has direct realizations in optics, as well as in the mean-field model of atomic Bose-Einstein condensates (BECs) [37; 38], and in many other areas. In particular, the spatial-domain light propagation in bulk media is modelled by the effectively two-dimensional version of Equation (7), with two transverse coordinates \((x,y)\):  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\left(\frac{\partial^{2}u}{\partial x ^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right)-U(x,y)u+\sigma|u|^{2}u=0. \tag{11}\]Unlike its one-dimensional counterpart (7), Equation (11) in the free space (\(U\left(x,y\right)=0\)) is not integrable.