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We prove that the flow of the discrete nonlinear Schrödinger 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    $$L^p$$  L p -estimate, for any    $$p \ge 1$$  p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.

Mean Field Derivation of DNLS from the Bose–Hubbard Model