  We present a non-equilibrium Green's functional approach to study the
dynamics following a quench in weakly interacting Bose Hubbard model (BHM). The
technique is based on the self-consistent solution of a set of equations which
represents a particular case of the most general set of Hedin's equations for
the interacting single-particle Green's function. We use the ladder
approximation as a skeleton diagram for the two-particle scattering amplitude
useful, through the self-energy in the Dyson equation, for finding the
interacting single-particle Green's function. This scheme is then implemented
numerically by a parallelized code. We exploit this approach to study the
correlation propagation after a quench in the interaction parameter, for one
(1D) and two (2D) dimensions. In particular, we show how our approach is able
to recover the crossover from ballistic to diffusive regime by increasing the
boson-boson interaction. Finally we also discuss the role of a thermal initial
state on the dynamics both for 1D and 2D Bose Hubbard models, finding that
surprisingly at high temperature a ballistic evolution is restored.
