  We present a framework of semiclassical superconductivity (SC) dynamics that
properly includes effects of spatial fluctuations for the attractive Hubbard
model. We consider both coherent and adiabatic limits. To model the coherent SC
dynamics, we develop a real-space von~Neumann equation based on the
time-dependent Hartree-Fock-Bogoliubov theory. Applying our method to
interaction quenches in the negative-$U$ Hubbard model, we show that the
relaxation of SC order at weak coupling is dominated by Landau-damping. At
strong coupling, we find a two-stage relaxation of the pairing field: a
collapse of the synchronized oscillation of Cooper pairs due to spatial
inhomogeneity, followed by a slow relaxation to a quasi-stationary state. SC
dynamics in adiabatic limit is described by a quantum Landau-Lifshitz equation
with Ginzburg-Landau relaxation. Numerical simulations of the pump-probe
process show that long time recovery of the pairing field is dominated by
defects dynamics. Our results demonstrate the important role of spatial
fluctuations in both limits.
