  Optical control and manipulation of cold atoms has become an important topic
in condensed matter. Widely employed are optical lattice shaking experiments
which allow the introduction of artificial gauge fields, the design of
topological bandstructures, and more general probing of quantum critical
phenomena. Here we develop new numerical methods to simulate these periodically
driven systems by implementing lattice shaking directly. As a result we avoid
the usual assumptions associated with a simplified picture based on Floquet
dynamics. A demonstrable success of our approach is that it yields quantitative
agreement with experiment, including Kibble-Zurek scaling. Importantly, we
argue that because their dynamics corresponds to an effective non-linear
Schr\"{o}dinger equation, these particular superfluid studies present a unique
opportunity to address how general Floquet band engineering is affected by
interactions. In particular, interactions cause instabilities at which the
behavior of the system changes dramatically.
