  We present a new variational method to study the dynamics of a closed bosonic
many-body system, the time-dependent hypernetted-chain Euler-Lagrange method,
tHNC . Based on the Jastrow ansatz, it accounts for quantum fluctuations in a
non-perturbative way. tHNC scales well with the number of dimensions, as
demonstrated by our results on one, two, and three dimensions. We apply the
tHNC method to interaction quenches, i.e. sudden changes of the interaction
strength, in homogeneous Bose gases. When the quench is strong enough that the
final state has roton excitations (as found and predicted for dipolar and
Rydberg-dressed Bose-Einstein condensates, respectively), the pair distribution
function exhibits stable oscillations. For validation, we compare tHNC results
with time-dependent variational Monte Carlo results in one and two dimensions.
