Fractional saturable impurity

Abstract

We examine analytically and numerically the effect of fractionality on a saturable bulk and surface impurity embedded in a one-dimensional lattice. We use a fractional Laplacian introduced previously by us, and by the use of lattice Green functions we are able to obtain the bound state energies and amplitude profiles, as a function of the fractional exponent $s$ and saturable impurity strength $\ensuremath{\chi}$ for both surface and bulk impurity. The transmission is obtained in closed form as a function of $s$ and $\ensuremath{\chi}$, showing strong deviations from the standard case, at small fractional exponent values. The self-trapping of an initially localized excitation is qualitatively similar for the bulk and surface mode, but in all cases complete confinement is obtained at $s\ensuremath{\rightarrow}0$, as shown theoretically and observed numerically.