  In this work, which is based on our previously derived theoretical framework
[1], we apply the truncated Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY)
hierarchy for ultracold bosonic systems with a fixed number of particles to two
out-of-equilibrium scenarios, namely tunneling in a double-well potential and
an interaction quench in a harmonic trap. The efficient formulation of the
theory provided in [1] allows for going to large truncation orders such that
the impact of the truncation order on the accuracy of the results can be
systematically explored. While the short-time dynamics is found to be
excellently described with controllable accuracy, significant deviations occur
on a longer time-scale for a sufficiently strong interaction quench or the
tunneling scenario. Theses deviations are accompanied by exponential-like
instabilities leading to unphysical results. The phenomenology of these
instabilities is investigated in detail and we show that the minimal-invasive
correction algorithm of the equation of motion as proposed in [1] can indeed
stabilize the BBGKY hierarchy truncated at the second order.
