Theory of space-charge waves on gradient-profile relativistic electron   beam: an analysis in propagating eigenmodes
  We developed an exact analytical treatment for space-charge waves within a relativistic electron beam in terms of (self-reproducing) propagating eigenmodes. This result is of obvious theoretical relevance as it constitutes one of the few exact solution for the evolution of charged particles under the action of self-interactions. It also has clear numerical applications in particle accelerator physics where it can be used as the first self-consistent benchmark for space-charge simulation programs. Today our work is of practical relevance in FEL technology in relation with all those schemes where an optically modulated electron beam is needed and with the study of longitudinal space-charge instabilities in magnetic bunch compressors. 
Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller and Mikhail   Yurkov