Soliton and rogue wave solutions of the space–time fractional nonlinear Schrödinger equation with [formula omitted]-symmetric and time-dependent potentials

Abstract

We derive fractional soliton and rogue wave solutions of the space–time fractional nonlinear Schrödinger (FNLS) equation in the existence of complex parity reflection–time reversal (PT)−symmetric and time-dependent potentials. We find that the fractional derivative variable transformation is a good approximation to reduce the space–time FNLS equation into its conventional counterpart. Utilizing the localized solutions of the reduced conventional system, we construct the fractional localized solutions for the system under consideration. We study the dynamics of localized fractional solitons with PT-symmetric Rosen–Morse, Scarff-II and time dependent potentials. We explore the impact of dynamical properties on the constructed fractional solitons by changing time, space fractional-order parameter, real and imaginary components of potential strengths. Our observations reveal that the soliton and RW profiles get distorted for low values of time and space fractional-order parameters, and they show the usual features when these parameters are close to unity. Furthermore, we also report that the intensity of localized profiles increases by varying the strength of the real part of potentials, and the unstable behaviour is exhibited for higher strengths of the imaginary part of potentials.