Thermal wave model for nonlinear longitudinal dynamics of a relativistic charged particle bunch in cold plasmas

We study the longitudinal dynamics of a relativistic charged particle bunch through a cold, unmagnetized plasma, within the framework of the recently proposed thermal wave model for relativistic charged particle beam propanation. We show that, under the action of both a purely electrostatic plasma wave potential well and the plasma wake potential (self-interaction), the longitudinal bunch dynamics is governed by a nonlinear Schrodinger equation for a complex wave function whose squared modulus is proportional to the longitudinal bunch density. This wave model, for which the diffraction parameter is represented here by the longitudinal emittance, allows us to study synchrotron-like oscillations in the plasma wave potential well as well as to obtain a longitudinal envelope equation which includes the self-interaction. Furthermore, we show that a soliton solution for the bunch density is possible and it results to be the natural asymptotic evolution of a modulational instability occurring when the bunch propagates in the plasma under the action of the self-force only.