Localized modes in nonlinear fractional systems with deep lattices

TL;DR

This paper investigates the existence, shapes, and stability of localized solitons, including vortex types, in deep lattice potentials within nonlinear fractional systems, using numerical and analytical methods.

Contribution

It provides new insights into the properties and stability of gap and vortex solitons in deep lattice fractional systems, including analytical approximations.

Findings

Fundamental gap solitons are tightly confined around a single lattice cell.

Vortex gap solitons are constructed with specific geometric configurations.

Stability of various localized modes is verified through linearization and simulations.

Abstract

Solitons in the fractional space, supported by lattice potentials, have recently attracted much interest. We consider the limit of deep one- and two-dimensional (1D and 2D) lattices in this system, featuring finite bandgaps separated by nearly flat Bloch bands. Such spectra are also a subject of great interest in current studies. The existence, shapes, and stability of various localized modes, including fundamental gap and vortex solitons, are investigated by means of numerical methods; some results are also obtained with the help of analytical approximations. In particular, the 1D and 2D gap solitons, belonging to the first and second finite bandgaps, are tightly confined around a single cell of the deep lattice. Vortex gap solitons are constructed as four-peak \textquotedblleft squares" and \textquotedblleft rhombuses" with imprinted winding number $S=1$. Stability of the solitons is explored by means of the linearization and verified by direct simulations.