Petviashvili Method for the Fractional Schr\"{o}dinger Equation

TL;DR

This paper extends the Petviashvili method to fractional nonlinear Schrödinger equations, enabling the construction and analysis of soliton solutions, and investigates their dynamics and interactions with various potentials.

Contribution

The paper introduces an extension of the Petviashvili method for fractional Schrödinger equations and analyzes soliton stability and interactions using spectral and Runge-Kutta methods.

Findings

Soliton solutions exhibit splitting and spreading behaviors.

The properties of solitons depend on the fractional order .

Potentials influence soliton dynamics and stability.

Abstract

In this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schr\"{o}dinger equation (fNLSE) for the construction and analysis of its soliton solutions. We also investigate the temporal dynamics and stabilities of the soliton solutions of the fNLSE by implementing a spectral method, in which the fractional-order spectral derivatives are computed using FFT routines, and the time integration is performed by a $4^{th}$ order Runge-Kutta time-stepping algorithm. We discuss the effects of the order of the fractional derivative, $\alpha$, on the properties, shapes, and temporal dynamics of the solitons solutions of the fNLSE. We also examine the interaction of those soliton solutions with zero, photorefractive and q-deformed Rosen-Morse potentials. We show that for all of these potentials the soliton solutions of the fNLSE exhibit a splitting and spreading behavior, yet their dynamics can be altered by the different forms of the potentials and noise considered.