Suppression of soliton collapses, modulational instability and rogue-wave excitation in two-Lévy-index fractional Kerr media

We introduce a generalized fractional nonlinear Schrödinger (FNLS) equation for the propagation of optical pulses in laser systems with two fractional-dispersion/diffraction terms, quantified by their Lévy indices, α 1   α 2 ∈ ( 1 , 2 ] , and self-focusing or defocusing Kerr nonlinearity. Some fundamental solitons are obtained by means of the variational approximation, which are verified by comparison with numerical results. We find that the soliton collapse, exhibited by the one-dimensional cubic FNLS equation with only one Lévy index (LI) α = 1 , can be suppressed in the two-LI FNLS system. Stability of the solitons is also explored against collisions with Gaussian pulses and adiabatic variation of the system parameters. Modulation instability (MI) of continuous waves is investigated in the two-LI system too. In particular, the MI may occur in the case of the defocusing nonlinearity when two diffraction coefficients have opposite signs. Using results for the MI, we produce first- and second-order rogue waves on top of continuous waves, for both signs of the Kerr nonlinearity.