## 1 An Overview of the Topic and Its Ramifications
### Dissipative and Parity-Time Symmetric Waveguides  
The brief discussion of the waveguiding mechanisms given above did not address the presence of losses and the necessity of compensating them by gain. This assumption is valid for relatively short propagation distances, as well as in the case when the compensating gain matches the action of losses so accurately that both factors may be simultaneously neglected in the first approximation. In reality, losses are an inevitably existing gradient in plasmonics and metamaterials, as the respective waveguides are based on metallic elements, which introduce the Ohmic dissipation.  
Generally speaking, if the medium is essentially lossy, the above-mentioned _index-guiding_ and _bandgap-guiding_ mechanisms which define the guiding channel(s), respectively, in terms of a transverse profile of the local refractive index, or the transmission-band structure induced by the PhC or PhC fiber may be replaced by a _gain-guiding scheme_ in which the signal propagates in a lossy planar or bulk medium along a narrow stripe of gain locally embedded into the medium [95; 96; 97].  
A recently developed topic which is closely related to the light transmission in dissipative waveguides deals with _the parity-time_ (\(\mathcal{PT}\)) _symmetry_, which implies balance between symmetrically (in space) placed gain and loss elements. A paradigmatic model (it often includes nonlinearity, although the \(\mathcal{PT}\) symmetry is by itself a linear property) is represented by NLSE (7), in which the potential is made _complex_, with real and imaginary parts being, respectively, spatially even and odd ones:  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{ \partial x^{2}}-\left[U_{r}(x)+i\mathit{l}\mathit{l}_{i}(x)\right]u+\sigma|u|^ {2}u=0,\] \[U_{r}(-x)=U_{r}(x),\;U_{i}(-x_{-}=-U_{i}(x). \tag{15}\]  
Another fundamental realization of the \(\mathcal{PT}\) symmetry in optics and related fields is offered by a coupler, in which one core carries uniformly distributed gain, and the parallel-coupled one is uniformly lossy, the accordingly modified Equation (8) being  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{ \partial x^{2}}+\kappa v-U(x)u+\sigma|u|^{2}u = i\gamma u, \tag{16}\] \[i\frac{\partial v}{\partial z}+\frac{1}{2}\frac{\partial^{2}v}{ \partial x^{2}}+\kappa u-U(x)v+\sigma|v|^{2}v = -i\gamma v,\]  
where \(\gamma>0\) is the gain-loss coefficient. The \(\mathcal{PT}\) symmetry has been experimentally realized in photonics, and a large number of guided-wave-propagation regimes have been investigated in such systems [98; 99; 100]. In particular, as concerns solitons, although \(\mathcal{PT}\)-symmetric systems belong to the class of dissipative ones, where solitons generally exist as isolated _attractors_, selected by the condition of the double balance between the dispersion (or diffraction) and nonlinearity, and between the gain and loss (the latter principle is very important for the creation of stable temporal solitons in fiber lasers [102]), in \(\mathcal{PT}\)-symmetric systems solitons exist in _continuous families_, similar to their counterparts in conservative models [100; 101]. In addition to the interest to fundamental studies, systems with the \(\mathcal{PT}\) symmetry offer promising applications, such as \"light diodes\", admitting unidirectional propagation of light in the waveguide, and lasers operating in the \(\mathcal{PT}\)-symmetric regime [103].