Mean Field Derivation of DNLS from the Bose-Hubbard Model

We prove that the flow of the discrete nonlinear Schrodinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose-Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative L-p-estimate, for any p >= 1, is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.