Fractional Discrete Linear and Nonlinear Models

Fractionality in physics is an old concept that has resurfaced recently, where the usual derivative and integral of integer order are extended to a fractional one. In this chapter, we exhibit several case studies of the effect of fractionality on the physics of some linear and nonlinear discrete systems. These systems include the discrete nonlinear Schrödinger equation in 1D and 2D, electrical transmission lines, linear and nonlinear impurities, and the Anderson model. In all of them, we examine the effect of replacing the usual Laplacian with its non-integer order version, characterized by a fractional exponent. In general, fractional effects seem to introduce a long-range coupling interaction among the sites of a lattice, a reduction of the bandwidth with decreasing exponent, an increased tendency towards degeneration, and a decrease of the threshold for the self-trapping transition in a nonlinear lattice. We also look at the interplay between fractionality and \(\mathcal{P}\mathcal{T}\) symmetry in some simple systems.