On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

Abstract

Abstract. We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.

Numerical Simulations of X-Ray Free Electron Lasers (XFEL)

“…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 [1].…”

“…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 [1].…”

“…In this paper, we propose the time-splitting (Bloch-decomposition based) pseudo-spectral methods to simulate the XFEL Schrödinger equation (with and without periodic potential). Our simulation results go far beyond known analytical results [1]. In particular we demonstrate numerically that the time-averaging procedure works in mass supercritical/energy subcritical cases, even up to blow-up time.…”

“…In [1] it was shown that, under appropriate assumptions, the solution of (1.1) can be approximated, in the asymptotic regime |ω| 1, by the solution of an effective, time-averaged Schrödinger initial value problem. The present work is a continuation of that line of investigation in more general contexts.…”

“…, as |ω| → ∞ (see [1] for more details). Passing back and forth between u and ψ (models (1.4) and (1.1)) is completely straightforward, by virtue of the transformation (1.2).…”