Transverse evolution of a long relativistic electron beam governed by the Vlasov-Poisson-type pair of equations within the plasma wake field dynamics in the local regime

An analysis of the transverse plasma wake field dynamics excited by a long relativistic electron beam is carried out in a purely local regime, in which the transverse beam spot-size is much bigger than the plasma wavelength. This is done by using the Vlasov equation, that governs the kinetic spatiotemporal evolution of the transverse paraxial beam transport. A self-consistent description of the beam dynamics is then provided by coupling the Vlasov equation with a Poisson-type equation. Furthermore, the latter relates the beam density with the plasma wake potential energy. Remarkably, the approach here proposed seems to be suitable for describing the beam self-modulation and the prediction of the beam collapse.