Petviashvili Method for the Fractional Schrödinger Equation

Abstract

In this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schrödinger 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 (Fast Fourier Transform) routines, and the time integration is performed by a 4th order Runge–Kutta time-stepping algorithm. We discuss the effects of the order of the fractional derivative, α, on the properties, shapes, and temporal dynamics of the soliton 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.

Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation

“…Recently, fractional order derivatives have been used in diverse real-life models of science and technology. Consequently, many researchers in fractional calculus have dedicated their devotion to recommending new fractional order derivatives, such as time fractional derivative [ 45 , 46 ], conformable space-time fractional [ 47 ], Riemann–Liouville fractional derivative[ 48 ], modified Riemann–Liouville fractional derivative [ 49 ], linear functional arguments using Chebyshev series [ 50 ], space–time fractional [ 51 , 52 ], Caputo derivative [ 53 , 54 ], etc.…”