  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.
