  A standard perturbative expansion around the mean-field solution is used to
derive the low-energy effective action for superconductors at T=0. Taking into
account the density fluctuations at the outset we get the effective action
where the density $\rho$ is the conjugated momentum to the phase $\theta$ of
the order parameter. In the hydrodynamic regime, the dynamics of the
superconductor is described by a time dependent non-linear Schr\"odinger
equation (TDNLS) for the field $\Psi(x)=\sqrt{\rho/2} e^{i\theta}$. The
evolution of the density fluctuations in the crossover from weak-coupling (BCS)
to strong-coupling (Bose condensation of localized pairs) superconductivity is
discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS
equation reduces to the the Gross-Pitaevskii equation for the order parameter,
as in the standard description of superfluidity. The conditions under which a
phase-only action can be derived in the presence of a long-range interaction to
describe the physics of the superconductivity of ``bad metals'' are discussed.
