## 1 An Overview of the Topic and Its Ramifications
### Nonlinear Waveguides
It admits formal soliton solutions, looked for as  
\[u\left(x,y;z\right)=\exp\left(ikz+iS\theta\right)U_{S}(r), \tag{12}\]  
in terms of the polar coordinates \(\left(r,\theta\right)\) in the \(\left(x,y\right)\) plane, where \(k>0\) is a real propagation constant, \(S=0,\pm 1,\pm 2\),..., is an integer _vorticity_ that may be embedded in the two-dimensional soliton (shaping it as a _vortex ring_), and \(U_{S}(r)\) is a real radial amplitude function satisfying boundary conditions \(U_{S}(r)\sim\exp\left(-\sqrt{2 kr}\right)\) at \(r\to\infty\), and \(U(r)\sim r^{\left|S\right|}\) at \(r\to 0\). Solitons (12) with \(S=0\) are often called _Townes solitons_[39]. However, the Townes solitons--as well as their vortex counterparts, with \(S\
eq 0\) in Equation (12)--are completely unstable, being vulnerable to destruction by the _critical collapse_ (formation of a singularity after a finite propagation distance) in the case of \(S=0\), and by a still stronger instability which splits vortex rings with \(S\
eq 0\)[39]. An important example of nonintegrable one-dimensional system modelling nonlinear light propagation in optics is the system of coupled-mode equations which describe the fiber Bragg gratings (i.e., nonlinear optical fibers with a periodic lattice of local defects permanently written in their cladding, with a period equal to half the wavelength of light coupled into this waveguide). The coupled-mode equations govern the evolution of amplitudes \(u\) and \(v\) of right- and left-traveling waves, which are mutually converted (reflected) into each other by the Bragg grating [40; 41]:  
\[ \begin{array}{rcl}iu_{t}+iu_{x}+\kappa v+\left(\frac{1}{2}|u|^{2}+|v|^{2} \right)u&=&0,\\ \\ iv_{t}-iv_{x}+\kappa u+\left(\frac{1}{2}|v|^{2}+|u|^{2}\right)v&=&0,\end{array}  \tag{13}\]  
where \(\kappa\) is the Bragg-grating reflectivity, and the group velocity of the light waves in the fiber is scaled to be 1. This system admits exact solutions in the form of solitons, but it is not an integrable one. Such solitons--moving in the fiber Bragg grating as in the waveguide--have been created in the experiment [42]. Roughly half of the soliton family is stable, and half unstable. The use of fiber Bragg gratings operating in the linear regime has grown into a large industry with many applications, such as sensors, dispersion compensators, optical buffers, etc. [43]. Another fundamentally important nonlinear model for the guided wave propagation is the one with the quadratic (alias second-harmonic) nonlinearity, instead of the cubic (Kerr) term in NLSE (7). The model is based on the propagation equations for complex amplitudes \(u\left(x,z\right)\) and \(v(x,z)\) of the fundamental and second harmonics [44; 45]:  
\[ \begin{array}{c}iu_{z}+\frac{1}{2}u_{xx}+vu^{*}=0,\\ \\ 2 iv_{z}-qv+\frac{1}{2}v_{xx}+\frac{1}{2}u^{2}=0,\end{array}  \tag{14}\]  
where \(q\) is a real mismatch parameter. Although it is a nonintegrable system, Equations (14) also give rise to solitons, which are generically found in a numerical form. These solitons form a family which is chiefly stable, with a small instability area [44; 45]. In BEC models, Equation (7), with evolution variable \(z\) replaced by (scaled) time, \(t\), is called the Gross-Pitaevskii equation, in which the cubic term represents--in the mean-field approximation--an average effect of collisions between atoms [37; 38]. The natural sign of the collision-induced term corresponds to self-repulsion (self-defocusing) (i.e., \(\sigma=-1\) in Equation (7)), but for atomic species such as \({}^{7}\)Li, \({}^{39}\)K, and \({}^{85}\)Rb, the sign may be switched to self-attraction by means of the Feshbach resonance, which is in turn controlled by a magnetic or laser field acting on the experimental setup [46]. Theoretical and experimental work with solitons and other diverse nonlinear effects (such as the modulational instability [47] and rogue waves [48; 49], shock waves, separation of immiscible components in binary systems, kinks, and domain walls [50], instantons [51], etc.)