Fractional Wave Models and Their Experimental Applications

A focused summary of one- and two-dimensional models for linear and nonlinear wave propagation in fractional media is given. The basic models, which represent fractional quantum mechanics and the emulation of paraxial fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, characterized by the respective Lévy indices. Basic species of one-dimensional fractional solitons, produced by the models which include cubic or quadratic nonlinear terms, are briefly outlined. In particular, it is demonstrated that the variational approximation works well in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber optics is presented too. In addition to the concise presentation of known results, perspectives for further work are outlined.