  We study the dynamics of self-trapping in Bose-Einstein condensates (BECs)
loaded in deep optical lattices with Gaussian initial conditions, when the
dynamics is well described by the Discrete Nonlinear Schr\"odinger Equation
(DNLS). In the literature an approximate dynamical phase diagram based on a
variational approach was introduced to distinguish different dynamical regimes:
diffusion, self-trapping and moving breathers. However, we find that the actual
DNLS dynamics shows a completely different diagram than the variational
prediction. We numerically calculate a detailed dynamical phase diagram
accurately describing the different dynamical regimes. It exhibits a complex
structure which can readily be tested in current experiments in BECs in optical
lattices and in optical waveguide arrays. Moreover, we derive an explicit
theoretical estimate for the transition to self-trapping in excellent agreement
with our numerical findings, which may be a valuable guide as well for future
studies on a quantum dynamical phase diagram based on the Bose-Hubbard
Hamiltonian.
