Comparing the full time-dependent Bogoliubov--de-Gennes equations to their linear approximation: A numerical investigation

In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen--Cooper--Schrieffer (BCS) equations, often also denoted as Bogoliubov--de--Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg--Landau-type of equation, in accordance with a recent theoretical results. We include a full description on the numerical implementation of the partial differential BCS\ BdG equations.