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

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.