We study the propagation of chirped $(D+1)$ -dimensional optical pulses in bulk media with periodic dispersion, analytically by using the variational approach and numerically by using a new, to our knowledge, numerical technique relying on the adaptive fast Hankel split-step method using cylindrical and spherical symmetries for two and three dimensions, respectively. Stability criteria for $(2+1)$ - and $(3+1)$ -dimensional solitons are identified, and the long-term dynamics of the solitons are studied with the averaged equations obtained using the Kapitza approach. Also, the slow dynamics of the solitons around the fixed points for the width and the chirp are studied. The importance of this research is in generating dispersion-managed optical solitons in optical communication. Also, this research is applied to the stabilization of the Bose–Einstein condensate in $(2+1)$ - and $(3+1)$ -dimensional optical lattices. We compare results of the new numerical technique with those obtained using the fast Fourier split-step technique.