Hermite-Gaussian–like soliton in the nonlocal nonlinear fractional Schrödinger equation

The propagation of an optical beam in the nonlocal nonlinear fractional Schrodinger equation (NNFSE) was numerically investigated. The results reveal that the stability of beam propagation decreases as the Levy index decreases in the local fractional case. However, the nonlocal response can effectively enhance the stability of beam propagation and leads to a completely stable solitons formation in the NNFSE. In addition, the shapes of fundamental mode solitons are determined by the Levy indexes and nonlocalities. For the high-order solitons case, the shapes slightly vary with the change of nonlocality, but are strongly dependent on the Levy index. Furthermore, the critical power of the stable soliton decreases as the Levy index decreases since the decrease of the Levy index will weaken the diffraction effect.