The Fractional Discrete Nonlinear Schr\"{o}dinger Equation

TL;DR

This paper introduces a fractional version of the discrete nonlinear Schrödinger equation, analyzing its spectral properties, nonlinear modes, and self-trapping behavior with long-range interactions governed by a fractional exponent.

Contribution

It develops the fractional discrete nonlinear Schrödinger equation, providing analytical and numerical insights into its spectrum, nonlinear modes, and self-trapping phenomena as functions of the fractional exponent.

Findings

Spectrum and mean square displacement computed in closed form.

Nonlinear modes and stability analyzed numerically.

Self-trapping transition shifts to lower thresholds with decreasing fractional exponent.

Abstract

We examine a fractional version of the discrete Nonlinear Schr\"{o}dinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero.