Accelerating Airy–Gauss–Kummer localized wave packets

•A general solution of 3D linear Schrödinger equation with an Airy time-dependence is reported.•We find that the Airy–Kummer spatiotemporal wave packets can carry infinite energy.•A class of the accelerating spatiotemporal optical pulses with special structures was found.•The spatiotemporal wave packets retain their energy features over several Rayleigh lengths.

A general approach to generating three-dimensional nondiffracting spatiotemporal solutions of the linear Schrödinger equation with an Airy-beam time-dependence is reported. A class of accelerating optical pulses with the structure of Airy–Gauss–Kummer vortex beams is obtained. Our results demonstrate that the optical field contributions to the Airy–Gauss–Kummer accelerating optical wave packets of the cylindrical symmetry can be characterized by the radial and angular mode numbers.