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The dynamics of Bose-Einstein condensates trapped in a deep optical latticeis governed by a discrete nonlinear equation (DNL). Its degree of nonlinearityand the intersite hopping rates are retrieved from a nonlinear tight-bindingapproximation taking into account the effective dimensionality of eachcondensate. We derive analytically the Bloch and the Bogoliubov excitationspectra, and the velocity of sound waves emitted by a traveling condensate.Within a Lagrangian formalism, we obtain Newtonian-like equations of motion oflocalized wavepackets. We calculate the ground-state atomic distribution in thepresence of an harmonic confining potential, and the frequencies of smallamplitude dipole and quadrupole oscillations. We finally quantize the DNL,recovering an extended Bose-Hubbard model

Nonlinear tight-binding approximation for Bose-Einstein condensates in a lattice