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We study the time-dependent Bogoliubov--de-Gennes equations for generictranslation-invariant fermionic many-body systems. For initial states that areclose to thermal equilibrium states at temperatures near the criticaltemperature, we show that the magnitude of the order parameter staysapproximately constant in time and, in particular, does not follow atime-dependent Ginzburg--Landau equation, which is often employed as aphenomenological description and predicts a decay of the order parameter intime. The full non-linear structure of the equations is necessary to understandthis behavior.

Incompatibility of time-dependent Bogoliubov--de-Gennes and Ginzburg--Landau equations