Stabilization of single- and multi-peak solitons in the fractional nonlinear Schroedinger equation with a trapping potential

TL;DR

This paper investigates the existence and stability of localized modes in the fractional nonlinear Schrödinger equation with a trapping potential, revealing stabilization mechanisms for various soliton structures under different nonlinearities and fractional orders.

Contribution

It provides approximate analytical solutions and stability analysis for localized modes in the fractional nonlinear Schrödinger equation with a trapping potential, including stabilization of higher-order modes.

Findings

Single-peak ground state stabilized at alpha=1

Dipole mode stabilized at alpha<1

Higher-order modes stabilized with quintic nonlinearity

Abstract

We address the existence and stability of localized modes in the framework of the fractional nonlinear Schroedinger equation (FNSE) with the focusing cubic or focusing-defocusing cubic-quintic nonlinearity and a confining harmonic-oscillator (HO) potential. Approximate analytical solutions are obtained in the form of Hermite-Gauss modes. The linear stability analysis and direct simulations reveal that, under the action of the cubic self-focusing, the single-peak ground state and dipole mode are stabilized by the HO potential at values of the Levy index (the fractionality degree) alpha = 1 and alpha < 1, which lead, respectively, to the critical or supercritical collapse in free space. In addition to that, the inclusion of the quintic self-defocusing provides stabilization of higher-order modes, with the number of local peaks up to seven, at least.