  A self-consistent nonlinear hydrodynamic theory is presented of the
propagation of a long and thin relativistic electron beam, for a typical plasma
wake field acceleration configuration in an unmagnetized and overdense plasma.
The random component of the trajectories of the beam particles as well as of
their velocity spread is modelled by an anisotropic temperature, allowing the
beam dynamics to be approximated as a 3-D adiabatic expansion/compression. {It
is shown that even in the absence of the {nonlinear plasma wake force}, the
localization of the beam in the transverse direction can be achieved owing to
the} nonlinearity associated with the adiabatic compression/rarefaction and a
coherent stationary state is constructed. Numerical calculations reveal the
possibility of the beam focussing and defocussing, but the lifetime of the beam
can be significantly extended by the appropriate adjustments, so that
transverse oscillations are observed, {similar to those} predicted within the
thermal wave and Vlasov kinetic models.
