  A self-consistent nonlinear hydrodynamic theory is presented of the
propagation of a long and thin relativistic electron beam through a plasma that
is relatively strongly magnetized, $|\Omega_e|\sim\omega_{pe}$ and whose
density is much bigger than that of the beam. In the regime when the parallel
phase velocity in the comoving frame is much smaller than the thermal speed and
the beam electrons are thermalized, a stationary solution for the beam is found
when the electron motion in the transverse direction is negligibile and the
transverse localization comes from the nonlinearity of its 3-D adiabatic
expansion. Conversely, when the parallel phase velocity is sufficiently large
to prevent the heat convection along the magnetic field, a helicoidally shaped
stationary beam is found whose transverse profile is determined from a
nonlinear dispersion relation and depends on the transverse size of the beam
and its pitch angle.
