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.

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In this paper, we provide analytical solutions describing the dynamic behavior of the Pearcey-Gaussian beams propagating in free space. Based on the analytical solutions, explicit expressions governing the focusing distances of the Pearcey-Gaussian beams are found and verified by numerical simulations. For the linearly chirped Pearcey-Gaussian beam, it exhibits a uni-focusing behavior during propagation. Particularly, the focusing distance is independent on the linear chirp parameter and remains z f  = 2 unchanged. Of particular interest is that the quadratically chirped Pearcey-Gaussian beam focuses twice when the quadratic chirp parameter β < 0. The first and the second focusing distances are determined by zf1 = 2/(1 − 4β) and zf2 = −1/(2β), respectively. Furthermore, we numerically investigate the peak powers at the different focusing positions and find that as β increases, the peak powers at zf1 and zf2 linearly decrease. It is expected that the characteristics can be used for manipulating the focusing distances and the peak powers to generate an optical beam with high peak power by adjusting the chirp parameter β.