Resonant propagation of x-rays from the linear to the nonlinear regime

We present a theoretical study of temporal, spectral, and spatial reshaping of intense, ultrafast x-ray pulses propagating through a resonant medium. Our calculations are based on the solution of a 3D time-dependent Schr\"odinger-Maxwell equation, with the incident x-ray photon energy on resonance with the core-level 1s-3p transition in neon. We study the evolution of the combined incident and medium-generated field, including the effects of stimulated emission, absorption, ionization and Auger decay, as a function of the input pulse energy and duration. We find that stimulated Raman scattering between core-excited states $1s^{-1}3p$ and $2p^{-1}3p$ occurs at high x-ray intensity, and that the emission around this frequency is strongly enhanced when also including the similar $1s^{-1}-2p^{-1}$ response of the ion. We also explore the dependence of x-ray self-induced transparency (SIT) and self-focusing on the pulse intensity and duration, and we find that the stimulated Raman scattering plays an important role in both effects. Finally, we discuss how these nonlinear effects may potentially be exploited as control parameters for pulse properties of x-ray free-electron laser sources.