We have investigated the propagation dynamics of super-Gaussian optical beams in fractional Schrödinger equation. We have identified the difference between the propagation dynamics of super-Gaussian beams and that of Gaussian beams. We show that, the linear propagation dynamics of the super-Gaussian beams with order m > 1 undergo an initial compression phase before they split into two sub-beams. The sub-beams with saddle shape separate each other and their interval increases linearly with propagation distance. In the nonlinear regime, the super-Gaussian beams evolve to become a single soliton, breathing soliton or soliton pair depending on the order of super-Gaussian beams, nonlinearity, as well as the Lévy index. In two dimensions, the linear evolution of super-Gaussian beams is similar to that for one dimension case, but the initial compression of the input super-Gaussian beams and the diffraction of the splitting beams are much stronger than that for one dimension case. While the nonlinear propagation of the super-Gaussian beams becomes much more unstable compared with that for the case of one dimension. Our results show the nonlinear effects can be tuned by varying the Lévy index in the fractional Schrödinger equation for a fixed input power.

We report on the observation of bound states of gain-guided solitary pulses (GGSPs) in a dispersion-managed erbium-doped fiber laser. Despite the fact that the GGSP is a chirped pulse and there is strong pulse stretching and compression along the cavity in the laser, the bound GGSPs observed have a fixed pulse separation, which is invariant to the pump strength change. Numerical simulation confirmed the experimental observations and further showed that not only the pulse separation but also the relative phase difference between the bound GGSPs remained fixed along the cavity.

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