  Understanding non-equilibrium quantum dynamics of many-body systems is one of
the most challenging problems in modern theoretical physics. While numerous
approximate and exact solutions exist for systems in equilibrium, examples of
non-equilibrium dynamics of many-body systems, which allow reliable theoretical
analysis, are few and far between. In this paper we discuss a broad class of
time-dependent interacting systems subject to external linear and parabolic
potentials, for which the many-body Schr\"{o}dinger equation can be solved
using a scaling transformation. We demonstrate that scaling solutions exist for
both local and nonlocal interactions and derive appropriate self-consistency
equations. We apply this approach to several specific experimentally relevant
examples of interacting bosons in one and two dimensions. As an intriguing
result we find that weakly and strongly interacting Bose-gases expanding from a
parabolic trap can exhibit very similar dynamics.
