Dynamics of discrete solitons in the fractional discrete nonlinear Schr\"odinger equation with the quasi-Riesz derivative

TL;DR

This paper introduces a novel fractional discrete nonlinear Schrödinger equation incorporating long-range interactions via a modified Riesz derivative, analyzes its linear spectrum, and explores the formation, stability, mobility, and collisions of discrete solitons.

Contribution

It presents the first study of a fractional discrete nonlinear Schrödinger equation with long-range interactions based on a modified Riesz derivative, including analytical and numerical analysis of soliton dynamics.

Findings

Exact dispersion relation for all Levy index values.

Existence and stability analysis of single- and two-site solitons.

Demonstration of soliton mobility and collision dynamics.

Abstract

We elaborate a fractional discrete nonlinear Schr\"{o}dinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its L\'{e}vy index (LI). This FDNLS equation represents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wavenumbers. Formation of single-site and two-site discrete solitons is explored, starting from the anti-continuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.