Mean-field quantum dynamics for a mixture of Bose-Einstein condensates

We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schr\"{o}dinger dynamics is effectively described by a system of coupled cubic non-linear Schr\"{o}dinger equations, one for each component.