Bright Solitons in Non-Hermitian systems: Exploring the interplay of $\mathcal{PT}$ Symmetry and Wadati Structure

In this study, we revisit the subject of optical solitons in non-Hermitian potentials. For the underlying model of the focusing nonlinear Schrödinger equation with a complex localized potential, we offer a classification of such models into four canonical cases. These correspond to (a) $\mathcal{PT}$-symmetric Wadati potentials, (b) $\mathcal{PT}$-symmetric non-Wadati potentials, (c) Non-$\mathcal{PT}$ but Wadati potentials, and (d) complex potentials with neither $\mathcal{PT}$ nor Wadati symmetries. We study the existence and stability of optical solitons across prototypical model variations spanning all four relevant quadrants. We complement such an investigation with a low-dimensional effective description based on a non-conservative variational approximation method. The variational framework is assessed in terms of its ability to represent the structural properties and bifurcations of solutions. {Our results suggest that continuous families of bright solitons persist across sectors (a)--(c). For non-$\mathcal{PT}$-symmetric potentials, small departures from the Wadati limit, corresponding to sector (d), suggest the loss of continuous families of bright-solitons including, more specifically, the possibility of the generic absence of single-humped localized states. The corresponding variational analysis is broadly consistent with these observations. We also highlight the usefulness and limitations for such variational approaches in describing bright solitons in the presence of non-Hermitian potentials.