Dynamics of three-dimensional localized wave packets defined by the spherical Bessel function $\mathbf{j}_0(x)$

We consider a localized wave packet which is an asymptotic solution to the three-dimensional wave equation and is generated by the spherical Bessel function $\mathbf{j}_0$ at the initial time. We discuss an approach for constructing an effective representation for such a packet in terms of special functions, based on the theory of the Maslov canonical operator and the study of the dynamics of the corresponding three-dimensional Lagrangian manifold. In particular, we prove that if the initial packet has a large momentum directed along the $x_3$ axis, then the Bessel structure eventually vanishes and the packet spreads. Specifically, on the corresponding Lagrangian manifold, there arise singularities of the fold and cusp types, which are associated with the Airy and Pearcey functions, respectively, as well as non-generic singularities corresponding to the asymptotics in terms of the error function.