Nonlinear behavior of the longitudinal modes of the coasting beam in a storage ring

A simple nonlinear model of a coasting beam coupled to a sharp storage ring-impedance is studied in the framework of the Vlasov equation. In the case of a Gaussian beam this non-perturbative formalism simplifies to a pair of equations of motion which together with the dispersion relation fully describe nonlinear saturation of initially unstable coherent mode. This, in turn, provides a stabilizing mechanism (via Landau damping) for the overall distribution function. Finally, some predictions about the energy overshoot are made