In this paper we use a similarity transformation connecting some families of
Nonlinear Schrodinger equations with time-varying coefficients with the
autonomous cubic nonlinear Schrodinger equation. This transformation allows one
to apply all known results for that equation to the non-autonomous case with
the additional dynamics introduced by the transformation itself. In particular,
using stationary solutions of the autonomous nonlinear Schrodinger equation we
can construct exact breathing solutions to multidimensional non-autonomous
nonlinear Schrodinger equations. An application is given in which we explicitly
construct time dependent coefficients leading to solutions displaying weak
collapse in three-dimensional scenarios. Our results can find physical
applicability in mean field models of Bose-Einstein condensates and in the
field of dispersion-managed optical systems