| Summary: | Accelerating beams are wave packets that appear to spontaneously accelerate without external potentials or applied forces. Since their first physical realization in the form of Airy beams, they have found applications on various platforms, spanning from optics to plasma physics. We investigate the dynamics of examples of finite-energy accelerating beams derived from catastrophe theory. We use a Madelung transformation in momentum space, combined with a wavelet transform analysis, to demonstrate that the beams' properties arise from a type of vanishing self-interference. We identify the modes responsible for the wave packet's acceleration, and we derive the general acceleration for higher-order cupsoid-related beams. We also demonstrate how bright solitons resulting from nonlinear Airy beams can be unambiguously detected using the wavelet transform. This methodology will allow for a better understanding of special wave packet dynamics.
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