  Thermal Wave Model is used to study the strong self-consistent Plasma Wake
Field interaction (transverse effects) between a strongly magnetized plasma and
a relativistic electron/positron beam travelling along the external magnetic
field, in the long beam limit, in terms of a nonlocal NLS equation and the
virial equation. In the linear regime, vortices predicted in terms of
Laguerre-Gauss beams characterized by non-zero orbital angular momentum (vortex
charge). In the nonlinear regime, criteria for collapse and stable oscillations
is established and the thin plasma lens mechanism is investigated, for beam
size much greater than the plasma wavelength. The beam squeezing and the
self-pinching equilibrium is predicted, for beam size much smaller than the
plasma wavelength, taking the aberrationless solution of the nonlocal Nonlinear
Schroeding equation.
