In all traditional techniques tracking the solution of the nonlinear Schroedinger (NLS) equation in (3+1) dimensions with spherical symmetry is time consuming because the problem in hand is (3+1) dimensional. In this work we study the propagation of chirped (3+1)-dimensional optical pulses in bulk media with periodic dispersion or alternating nonlinearity, analytically by using the variational approach, and numerically by using a new numerical technique relying on the adaptive spherical Fourier Bessel split-step (ASFBSS) method using spherical symmetry for 3 dimensions respectively. Using fast algorithms for spherical Fourier Bessel transforms along with adaptive longitudinal stepping and transverse grid management in a symmetrized split-step technique, it is possible to accurately study many nonlinear effects, including the possibility of spatio-temporal collapse, or the collapse-arresting mechanism due to saturable nonlinearity dispersion management or alternating nonlinearity. Stability criteria for (3+1)- dimensional solitons are identified, and the long term dynamics of the solitons are studied by 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 work is in generating dispersion managed or nonlinearly managed optical solitons in optical communication. Also, this work is applied to the stabilization of the Bose-Einstein condensate in (3+1)- dimensional optical lattice. We compare results of the new numerical technique with those obtained using fast Fourier split step (FFSS) technique. We feel that the ASFBSS is a fast and more accurate method for tracking spatiotemporal pulse propagation in a saturated and un saturated nonlinear Kerr medium in which the spatiotemporal collapse is expected for the paraxial unsaturated medium approximation, and to cyclic focusing and defocusing in saturable, dispersion managed or alternating non-linearity media.