  We study the time-dependent Bogoliubov--de-Gennes equations for generic
translation-invariant fermionic many-body systems. For initial states that are
close to thermal equilibrium states at temperatures near the critical
temperature, we show that the magnitude of the order parameter stays
approximately constant in time and, in particular, does not follow a
time-dependent Ginzburg--Landau equation, which is often employed as a
phenomenological description and predicts a decay of the order parameter in
time. The full non-linear structure of the equations is necessary to understand
this behavior.
