Self-modulation of a relativistic charged-particle beam as thermal matter wave envelope

The self-modulation, resulting from its interaction with the surrounding medium, of a relativistic charged-particle beam traveling through an overdense plasma, is investigated theoretically. The description of the transverse nonlinear and collective beam dynamics of an electron (or positron) beam in a plasma-based accelerator is provided in terms of a thermal matter wave envelope propagation. This is done using the quantum-like description provided by the thermal wave model. It is shown that the charged-particle beam dynamics is governed by a Zakharov-type system of equations, comprising a nonlinear Schrödinger equation that is governing the spatiotemporal evolution of the thermal matter wave envelope and a Poisson-like equation for the wake potential that is generated by the bunch itself