  A review of a recent theoretical investigation of the quantum transverse beam
motion that has been developed in terms of a coupled system of nonlinear
spinorial equations is carried out. This is done assuming that a relativistic
electron/positron beam is traveling through a plasma along an external magnetic
field in overdense condition. Three different nonliner regimes of this analysis
are considered: the strictly local regime, where the beam spot size is much
greater than the plasma wavelength (then, the spinorial system is reduced to a
2D cubic nonlinear Schrodinger equation in the form of Gross-Pitaevskii
equation); the moderately nonlocal regime where the beam spot size is
comparable to the plasma wavelength; and, finally, the strongly nonlocal regime
where the beam spot size is much smaller than the plasma wavelength. In all the
cases the existence of quantum beam vortices and ring solitons is demonstrated
both analytically and numerically.
