Radiation of accelerating pulses with specified envelopes

A problem for radiation of time accelerating electromagnetic pulses is considered in paraxial approximation. It is shown that the pulse envelope moves in the times-patial coordinates on the surface of a cylinder, the parabolic one for the pulse in the form of Airy function and for the hyperbolic one for Gaussian. Each of the pulse propagates in time with deceleration along the dominant propagation direction and drifts uniformly in the lateral direction stopping at infinity.