  The dynamics of Bose-Einstein condensates trapped in a deep optical lattice
is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity
and the intersite hopping rates are retrieved from a nonlinear tight-binding
approximation taking into account the effective dimensionality of each
condensate. We derive analytically the Bloch and the Bogoliubov excitation
spectra, and the velocity of sound waves emitted by a traveling condensate.
Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of
localized wavepackets. We calculate the ground-state atomic distribution in the
presence of an harmonic confining potential, and the frequencies of small
amplitude dipole and quadrupole oscillations. We finally quantize the DNL,
recovering an extended Bose-Hubbard model.
