Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity

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 α. All solitons are demonstrated to be stable for 1<α≤2. However, when α=1, the beam undergoes a catastrophic collapse (blow-up) like its counterpart in the (1+2)-dimensional system at α=2. The collapse distance is analytically obtained and a physical explanation for the collapse is given.