## 1 An Overview of the Topic and Its Ramifications
### Linear Waveguides  
The basic waveguiding structure is a single-mode conduit, designed with a sufficiently small transverse size and boundary conditions at the boundary between the guiding core and surrounding cladding, which admits the propagation of a single transverse mode, while allhigher-order modes get imaginary propagation constants (i.e., they cannot actually propagate). A commonly known--and arguably the most important--example is provided by single-mode optical fibers (although, strictly speaking, all such fibers are bimodal if the polarization of light is taken into consideration) [17; 18]. Single-mode waveguides are crucially important components of telecommunication systems, while other applications (e.g., the delivery of powerful laser beams for material processing and the creation of complex spatiotemporal patterns) are best served by multimode conduits [19; 20]. Parallel to waveguiding fibers, planar waveguides are a subject of many studies in optics. In the corresponding models, as well as in their fiber-optics counterparts, the evolution variable is the propagation distance, \(z\); see Equation (2) below (this is a common feature of all guided-wave-propagation settings, not only in optics, but in other physical realizations of waveguides as well). Meanwhile, the transverse coordinate, \(x\), in the _spatial domain_ plays the same role as the reduced-time variable,  
\[\tau\equiv t-V_{\rm gr}^{-1}z, \tag{1}\]  
where \(t\) is the time proper, and \(V_{\rm gr}\) is the group velocity of the carrier wave in the _temporal domain_ in fiber optics. The waveguiding structure in the planar waveguide is represented (roughly speaking) by a stripe with a locally increased effective refractive index. Effective equations which model the temporal-domain propagation of optical waves in fibers and the spatial-domain propagation in planar waveguides are similar to each other, taking the form of the linear Schrodinger equation for local amplitude \(u\) of the electromagnetic wave, which is written here in terms of the spatial-domain propagation, and in the scaled form:  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{\partial x^{2} }-U(x)u=0. \tag{2}\]  
In particular, the aforementioned stripe waveguiding channel is represented by trapping potential \(U(x)\) in Equation (2), while the second derivative in Equation (2) represents the paraxial (weak) transverse diffraction in the planar waveguide. A ubiquitous form of the potential is  
\[U(x)=-\epsilon\;{\rm sech}^{2}(x/l), \tag{3}\]  
where \(\epsilon>0\) determines the effective depth of the potential well, and \(l\) determines its width. In the temporal domain, the transverse coordinate, \(x\), is replaced by the above-mentioned temporal variable (1). and the diffraction term in Equation (2) is replaced by \(-(\beta/2)\partial^{2}u/\partial\tau^{2}\), where \(\beta\) is the coefficient of the group-velocity dispersion (\(\beta>0\) and \(\beta<0\) correspond to the normal and anomalous dispersion, respectively). Further, the similarity between the wave-propagation Equation (2) in optics and the Schrodinger equation in quantum mechanics suggests a similarity between the guided transmission of waves in the guiding channel and propagation of real quantum particles in holding channel potentials [21]. The consideration of the transport of quantum particles in such channels gives rise to many intriguing peculiarities, such as the consideration of curved guiding channels. In this context, it is relevant to mention a well-known result which demonstrates a strong effect of the confinement imposed by a pipe-shaped potential on the character of the effectively one-dimensional mutual scattering of two quantum particles, which amounts to full reflection of the colliding particles [22]. This theoretical prediction had suggested the experimental realization of the concept of the _Tonks-Girardeau gas_; i.e., a gas composed of _hard-core bosons_, which bounce back from each other when they collide [23; 24].