One-dimensional gap solitons in quintic and cubic-quintic Fractional nonlinear Schr\"{o}dinger equations with a periodically modulated linear potential

TL;DR

This paper explores the existence and stability of one-dimensional gap solitons in a fractional nonlinear Schrödinger equation with cubic-quintic nonlinearities and a periodic potential, revealing stable localized modes through stability analysis.

Contribution

It extends the study of gap solitons to fractional nonlinear Schrödinger equations with competing nonlinearities and a periodic potential, identifying stability conditions for localized modes.

Findings

Identified fundamental and dipole-mode gap solitons in the fractional model.

Determined stability regions using linear stability analysis and simulations.

Confirmed the anti-Vakhitov-Kolokolov criterion for stable solitons.

Abstract

Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schr\"{o}dinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years---the nonlinear fractional Schr\"{o}dinger equation (NLFSE), therefore obtaining the cubic-quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov-Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic-quintic nonlinear terms.