  We examine the linear and nonlinear modes of a one-dimensional nonlinear
electrical lattice, where the usual discrete Laplacian is replaced by a
fractional discrete Laplacian. This induces a long-range intersite coupling
that, at long distances, decreases as a power law. In the linear regime, we
compute both, the spectrum of plane waves and the mean square displacement
(MSD) of an initially localized excitation, in closed form in terms of
regularized hypergeometric functions and the fractional exponent. The MSD shows
ballistic behavior at long times, MSD$\sim t^2$ for all fractional exponents.
When the fractional exponent is decreased from its standard integer value, the
bandwidth decreases and the density of states shows a tendency towards
degeneracy. In the limit of a vanishing exponent, the system becomes completely
degenerate. For the nonlinear regime, we compute numerically the low-lying
nonlinear modes, as a function of the fractional exponent. A modulational
stability computation shows that, as the fractional exponent decreases, the
number of electrical discrete solitons generated also decreases, eventually
collapsing into a single soliton.
