Modeling of self-consistent distributions for longitudinally non-uniform beams

Self-consistent distributions for beams in longitudinal magnetic and electric fields are considered. The magnetic and electric fields are assumed to be uniform in the beam cross-section, but can slowly vary along the beam axis. As a consequence, the beam cross-section radius and the longitudinal velocities of particles can also vary in the longitudinal direction. New classes of self-consistent distributions are found as analytical solutions of the Vlasov equation. Some known distributions are particular cases of these distributions. The analytically described distributions were also numerically simulated.