Fractional-order effect on soliton wave conversion and stability for the two-Lévy-index fractional nonlinear Schrödinger equation with PT-symmetric potential

We investigate a variable-coefficient fractional nonlinear Schrödinger(vc-FNLS) equation with Wadati potential and PT-symmetric potential. We find the Lévy index can be used to transition from a breather wave to a soliton as the fractional order derivative is increasing. The influences of fractional α and β on the breather, dark and bright solitons of space–time vc-FNLS equation are analyzed in detail. Some different propagation dynamics are generated via the different parameters α and β. We find that the value of α decreases, the number of oscillations or singularities increases for small time values and decreases for large time values. And the stability of fundamental soliton and asymmetric soliton is explored via the linear stability analysis and direct propagations. Interestingly stability of the soliton is also explored against collisions with Wadati potential and PT-symmetric potential. To additionally explore robustness of the solitons in the vc-FNLS model, it is also relevant to show that they keep its integrity against “bombardment” by impinging waves. These states indicate that the interaction acts both repulsively and attractively, and the analysis can be extended for vc-FNLS equations with more than two different fractional-diffraction terms. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials.