Dynamics of Gaussian beam modeled by fractional Schrödinger equation with a variable coefficient

In the paper, we investigate the propagation dynamics of the Gaussian beam modeled by the fractional Schrödinger equation (FSE) with a variable coefficient. In the absence of the beam's chirp, for smaller Lévy index, the Gaussian beam firstly splits into two beams, however under the action of the longitudinal periodic modulation, they exhibit a periodically oscillating behaviour. And with the increasing of the Lévy index, the splitting behaviour gradually diminishes. Until the Lévy index equals to 2, the splitting behaviour is completely replaced by a periodic diffraction behaviour. In the presence of the beam's chirp, one of the splitting beams is gradually suppressed with the increasing of the chirp, while another beam on the opposite direction becomes stronger and exhibits a periodically oscillating behaviour. Also, the oscillating amplitude and period are investigated and the results show that the former is dependent on the modulation frequency, the Lévy index and the beam's chirp, the latter depends only on the modulation frequency. Thus, the evolution of the Gaussian beam can be well manipulated to achieve the beam management in the framework of the FSE by controlling the system parameters and the chirp parameter.