Quantum ring soliton formation by strongly nonlocal plasma wake field response to a relativistic electron beam

The relativistic electron/positron particle beam propagation in overdense magnetized plasmas is studied theoretically, using a fluid plasma model and accounting for the quantum properties of individual particles. The collective character of the particle beam manifests through the macroscopic, beam created, plasma wake field. The transverse dynamics is described by the quantum Schrödinger equation for the single-particle wave function, within the Hartree mean-field approximation, coupled with the Poisson equations for the wake potential. The resulting nonlinear nonlocal Schrödinger equation is solved analytically in the strongly nonlocal regime, yielding breathing/wiggling Hermite-Gauss ring solitons. The nonstationary rings may be parametrically unstable. The conditions for instability and the growth rates are estimated analytically.