Basic fractional nonlinear-wave models and solitons

This review article provides a concise summary of one- and two-dimensionalmodels for the propagation of linear and nonlinear waves in fractional media.The basic models, which originate from fractional quantum mechanics and moreexperimentally relevant setups emulating fractional diffraction in optics, arebased on the Riesz definition of fractional derivatives, which arecharacterized by the respective Levy indices. Basic species of one-dimensionalsolitons, produced by the fractional models which include cubic or quadraticnonlinear terms, are outlined too. In particular, it is demonstrated that thevariational approximation is relevant in many cases. A summary of the recentlydemonstrated experimental realization of the fractional group-velocitydispersion in fiber lasers is also presented.