The interplay between wave localisation and self-acceleration, makes radially self-accelerating beams to lose their self-accelerating character, in favour of a pseudo self-acceleration, which deforms their transverse intensity distribution. Read more about it in our latest publication:
Marco Ornigotti, and Alexander Szameit, “Radially self-accelerating optical pulses“, Phys. Rev. A 99, 023859 (2019).