  We study a nonlinear Schr\"odinger equation which arises as an effective
single particle model in X-ray Free Electron Lasers (XFEL). This equation
appears as a first-principles model for the beam-matter interactions that would
take place in an XFEL molecular imaging experiment in \cite{frat1}. Since XFEL
is more powerful by several orders of magnitude than more conventional lasers,
the systematic investigation of many of the standard assumptions and
approximations has attracted increased attention.
  In this model the electrons move under a rapidly oscillating electromagnetic
field, and the convergence of the problem to an effective time-averaged one is
examined. We use an operator splitting pseudo-spectral method to investigate
numerically the behaviour of the model versus its time-averaged version in
complex situations, namely the energy subcritical/mass supercritical case, and
in the presence of a periodic lattice.
  We find the time averaged model to be an effective approximation, even close
to blowup, for fast enough oscillations of the external field. This work
extends previous analytical results for simpler cases \cite{xfel1}.
