Self consistent thermal wave model description of the transverse   dynamics for relativistic charged particle beams in magnetoactive plasmas
  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. 
Renato Fedele, Fatema Tanjia, Sergio De Nicola, P. K. Shukla, and   Dusan Jovanovic
physics.plasm-ph physics.acc-ph