  Self modulated dynamics of a relativistic charged particle beam is reviewed
within the context of the theory of plasma wake field excitation. The
self-consistent description of the beam dynamics is provided by coupling the
Vlasov equation with a Poisson-type equation relating the plasma wake potential
to the beam density. An analysis of the beam envelope self-modulation is then
carried out and the criteria for the occurrence of the instability are
discussed thereby.
