  In this paper, we implement the Adaptive Moving Mesh method (AMM) to the
solution of initial value problems involving the Schr\"odinger equation, and
more specifically the Schr\"odinger-Poisson system of equations. This method is
based on the solution of the problem on a discrete domain, whose resolution is
coordinate and time-dependent, and allows to dynamically assign numerical
resolution in terms of desired refinement criteria. We apply the method to
solve various test problems involving stationary solutions of the SP system,
and toy scenarios related to the disruption of subhalo s made of ultralight
bosonic dark matter traveling on top of host galaxies.
