Kinetic theory of longitudinal stability analysis of a non-laminar electron beam in self-consistent plasma wake field excitation

We carry out a stability analysis of a relativistic nonlaminar electron beam which is experiencing the self-consistent plasma wake field excitation. This is done in overdense regime (i.e. plasma density much greater than beam density) in a cold plasma. We adopt the self-consistent kinetic model for the plasma wake field excitation, that is based on the pair of Vlasov-Poisson-type equations. The latter governs the phase-space spatiotemporal evolution of the beam and its linearized form leads to a Landau-type approach to the beam stability analysis. Thereby, the analysis, performed for the case of a Gaussian electron beam distribution, shows the existence of the unstable modes for both cold and warm beams, respectively.