Self-accelerating beam dynamics in the space fractional Schr\"odinger equation

Self-accelerating beams are fascinating solutions of the Schr\"odinger equation. Thanks to their particular phase engineering, they can accelerate without the need of external potentials or applied forces. Finite-energy approximations of these beams have led to many applications, spanning from particle manipulation to robust in vivo imaging. The most studied and emblematic beam, the Airy beam, has been recently investigated in the context of the fractional Schr\"odinger equation. It was notably found that the packet acceleration would decrease with the reduction of the fractional order. Here, I study the case of a general nth-order self-accelerating caustic beam in the fractional Schr\"odinger equation. Using a Madelung decomposition combined with the wavelet transform, I derive the analytical expression of the beam's acceleration. I show that the non-accelerating limit is reached for infinite phase order or when the fractional order is reduced to 1. This work provides a quantitative description of self-accelerating caustic beams' properties.