Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrodinger equation

TL;DR

This paper investigates the existence and orbital stability of periodic standing wave solutions for the cubic fractional nonlinear Schrödinger equation, combining analytical methods and numerical simulations.

Contribution

It introduces a new approach to establish stability using positive operators, oscillation theorems, and the Vakhitov-Kolokolov condition, along with numerical generation of solutions.

Findings

Existence of positive real solutions for the fNLS equation.

Orbital stability proven using combined analytical tools.

Numerical validation of solutions and Vakhitov-Kolokolov condition.

Abstract

In this paper, the existence and orbital stability of the periodic standing waves solutions for the nonlinear fractional Schrodinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and we it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding positive operators, the oscillation theorem for fractional Hill operators and a Vakhitov-Kolokolov condition, well known for Schrodinger equations. We then perform a numerical approach to generate periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.