Dynamics of solitons in Lugiato–Lefever cavities with fractional diffraction

We study soliton dynamics in a passive optical system governed by the Lugiato–Lefever equation with effective diffraction represented by a fractional spatial derivative. Two models are considered, with the spatially uniform or strongly localized pump. Stable (quasi-) solitons are constructed, and their stability regions in the systems' parameter spaced are identified primarily, in a numerical form. The stability is strongly affected by the Lévy index (LI) of the fractional diffraction. The dependence of the stability on the loss coefficient and pump strength, as well as the localization size in the case of the confined pump, are studied too. In the latter case, the stability is enhanced by the narrow localization. Unstable solitons spontaneously develop intrinsic oscillations. In the case of the localized pump, some unstable solitons escape from the position pinned to the pumped region.