  A numerical investigation on the spatiotemporal evolution of an electron
beam, externally injected in a plasma in the presence of a plasma wake field,
is carried out. The latter is driven by an ultra-short relativistic
axially-symmetric femtosecond electron bunch. We first derive a novel
Poisson-like equation for the wake potential where the driving term is the
ultra-short bunch density, taking suitably into account the interplay between
the sharpness and high energy of the bunch. Then, we show that a channel is
formed longitudinally, through the externally injected beam while experiencing
the effects of the bunch-driven plasma wake field, within the context of
thermal wave model. The formation of the channel seems to be a final stage of
the 3D evolution of the beam. This involves the appearance of small filaments
and bubbles around the longitudinal axis. The bubbles coalesce forming a
relatively stable axially-symmetric hollow beam structure.
