We investigate the nonlinear dynamics of (1+1)-dimensional optical beam in the system described by the space-fractional Schrödinger equation with the Kerr nonlinearity. Using the variational method, the analytical soliton solutions are obtained for different values of the fractional Lévy index $\alpha$. All solitons are demonstrated to be stable for $1<\alpha \le 2$. However, when $\alpha =1$, the beam undergoes a catastrophic collapse (blow-up) like its counterpart in the (1+2)-dimensional system at $\alpha =2$. The collapse distance is analytically obtained and a physical explanation for the collapse is given.

The fractional Fourier transform (FRFT) naturally exists in strongly nonlocal nonlinear (SNN) media and the propagation of optical beams in SNN media can be simply regarded as a self-induced FRFT. Through the FRFT technique the evolution of fields in SNN media can be conveniently dealt with, and an arbitrary square-integrable input field presents itself generally as a revivable higher-order spatial soliton which reconstructs its profile periodically after every four Fourier transforms. The self-induced FRFT illuminates the prospects for SNN media in new applications such as continuously tunable nonlinearity-induced FRFT devices.

The existence and stability of fundamental, dipole, and tripole solitons in Kerr nonlinear media with parity-time-symmetric Gaussian complex potentials are reported. Fundamental solitons are stable not only in deep potentials but also in shallow potentials. Dipole and tripole solitons are stable only in deep potentials, and tripole solitons are stable in deeper potentials than for dipole solitons. The stable regions of solitons increase with increasing potential depth. The power of solitons increases with increasing propagation constant or decreasing modulation depth of the potentials.

The existence and stability of defect solitons in parity-time (PT) symmetric optical lattices with nonlocal nonlinearity are reported. It is found that nonlocality can expand the stability region of defect solitons. For positive or zero defects, fundamental and dipole solitons can exist stably in the semi-infinite gap and the first gap, respectively. For negative defects, fundamental solitons can be stable in both the semi-infinite gap and the first gap, whereas dipole solitons are unstable in the first gap. There exist a maximum degree of nonlocal nonlinearity, above which the fundamental solitons in the semi-infinite gap and the dipole solitons in the first gap do not exist for negative defects. The influence of the imaginary part of the PT-symmetric potentials on soliton stability is given. When the modulation depth of the PT-symmetric lattices is small, defect solitons can be stable for positive and zero defects, even if the PT-symmetric potential is above the phase transition point.

We report, to the best of our knowledge, the first experimental observation of higher-charge vortex solitons and vector vortex solitons in lead glass with strongly thermal nonlocal nonlinearity. A higher-charge vortex soliton with a topological charge of $l=4$ and a vector vortex soliton consisting of two orthogonally polarized vortex components, with charges $l_1=1$ and $l_2=4$, were observed at several times of diffraction length. We show that the ring profiles and the carried topological charges of the two incoherently coupled vortex components can be preserved. We also numerically find that the stability of the higher-charge vortex can be enhanced by co-propagating a stable, single-charge vortex.

We investigate the propagation of spatial solitons in cylindrical strongly nonlocal media by a method of image beam of light. The dynamic force of the soliton steering resulting from the boundary effect is equivalent to the force between the soliton beam and the image beam. The trajectory of the soliton is analytically studied, which is in good agreement with the experimental results.