  We study the time evolution of bosonic systems where multiple driven bosonic
modes of light interact with multiple mechanical resonators through arbitrary,
time-dependent, optomechanical-like interactions. We find the analytical
expression for the full time evolution of the system and compute the
expectation value of relevant quantities of interest. Among the most
interesting ones, we are able to compute the first-order quantum bipartite
coherence between pairs of subsystems, and the analytical expression for the
mixedness induced by the nonlinear interaction in the reduced state of the
mechanical oscillators. Finally, we also compare our results with a linearised
version of the system, and we find a regime where there are qualitative and
quantitative differences in the behavior of some measurable quantities. Our
results can therefore be used to describe the full time-evolution of the
system, to characterise its nonlinear character and explore the validity of the
linearisation approach.
