Solitons in spin-orbit-coupled systems with fractional spatial derivatives

We demonstrate the existence of various types of solitons in the spin-orbit-coupled systems with the fractional dimension based on Lévy random flights, including the systems with or without Zeeman splitting. Specifically, the systems without Zeeman splitting can support families of symmetric solitons, whereas the systems with Zeeman splitting can support families of stable asymmetric solitons. These coupled solitons may come in the form of fundamental single solitons or dipole solitons. The Lévy index, the strength of self- and cross-phase modulation, and the propagation constant strongly affect the waveforms and stability domains of coupled solitons. The stability and instability domains of such single and dipole solitons are calculated by the method of linear stability analysis and are confirmed by the numerical simulation of perturbed propagation. The general conclusion is that for the Lévy index close to 2, corresponding to the normal nonlinear optics, the solitons tend to be stable, while in the opposite case of Lévy index close to 1, corresponding to Cauchy random flights, the solitons tend to become unstable.