Nonlinear saturation 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 formulated in the framework of the quasilinear Vlasov equation. Nonperturbative analytic treatment of the Vlasov equation allows us to study time evolution of a single coherent mode (an azimuthal harmonic of the density driven by the impedance) and the overall uniform-density distribution function. In the case of a Gaussian beam, this formalism simplifies to a pair of equations of motion which together with the dispersion relation fully describe the dynamics of the beam. Further numerical treatment reveals saturation of the mode growth which simultaneously provides a stabilizing mechanism (via Landau damping) for the overall distribution function. Some predictions about the energy overshoot and coherent-instability lifetime are made on the basis of the presented formalism.