Fundamental, dipole, and vortex solitons in fractional nonlinear Schrödinger equation with a parity-time-symmetric periodic potential

We investigate the existence and stability of two-dimensional (2D) solitons in the framework of the nonlinear fractional Schrödinger equation (NLFSE) with focusing nonlinearity and a parity-time (PT) symmetric periodic potential. Numerical studies reveal that the fundamental, dipole, and localized vortex solitons exist in a semi-infinite band gap. The Lévy index (α) cannot change the phase transition point of the PT-symmetric periodic potential, but it affects the existence and stability of these solitons significantly. With the growth of the Lévy index, the stability regions become markedly wider for the fundamental, dipole solitons and vortex solitons with the topological charge of 2 (S=2). But for a unit topological charge (S=1), the vortex soliton cannot maintain its stability under the usual diffraction effect (α=2). The result shows that the fractional-order diffraction helps to stabilize the vortex solitons with S=1.