Effects of fractional diffraction on nonlinear <i>P</i><i>T</i> phase transitions and stability of dark solitons and vortices

The wave propagation under the action of fractional diffraction has recently drawn increasing attention in nonlinear optics. Here, we address the effect of fractional diffraction on the existence, phase transitions, and stability of dark solitons (DSs) and vortices in parity-time (<i>P</i><i>T</i>) symmetric graded-index waveguide with self-defocusing nonlinearity. The DSs and vortices are produced by numerical solution of the corresponding one- and two-dimensional fractional nonlinear Schrödinger equations. We show that solution branches of fundamental and higher-order DSs collide pair-wise (merge) and disappear with the increase of the gain-loss strength, revealing nonlinear <i>P</i><i>T</i> phase transitions in the waveguide. Numerically identifying the merger points, we demonstrate the effects of the fractional diffraction on the phase transition. The phase transition points determine boundaries of the existing regions for the DSs and vortices. The stability of the DSs and vortices is studied using linearization for small perturbations. Direct simulations of perturbed evolution corroborate their stability properties predicted by the analysis of small perturbations.