Solitary Waves in an Intense Beam Propagating through a Smooth Focusing Field

Based on the Vlasov-Maxwell equations describing the self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. In an earlier paper [1] it has been shown that in the case of a beam with uniform phase space density the Vlasov-Maxwell equations can be replaced exactly by the macroscopic warm fluid-Maxwell equations with a triple adiabatic pressure law. In this paper we demonstrate that starting from the macroscopic fluid-Maxwell equations a nonlinear Schroedinger equation for the slowly varying wave amplitude (or a set of coupled nonlinear Schroedinger equations for the wave amplitudes in the case of multi-wave interactions) can be derived. Properties of the nonlinear Schroedinger equation are discussed, together with soliton formation in intense particle beams.