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

Numerical simulations of X-rays Free Electron Lasers (XFEL)