Saturation of a Longitudinal Instability Due to Nonlinearity of the Wake Field

Self-sustained synchrotron oscillations are observed in electron storage rings. In general the theoretical description of the saturation of an instability for large oscillation amplitude is a difficult problem, and techniques have not yet been developed which yield analytic approximations to the appropriate nonlinear Vlasov or Fokker-Planck equations. In this paper, a single point bunch interacting with the wake field from a single resonant mode of an RF cavity is considered, and the averaging method of Bogoliubov and Mitropolsky is used to study the saturation of the initial exponential growth of the oscillation amplitude, due to the nonlinearity of the wake field. The determination of the limiting amplitude of oscillation is discussed both in the presence and in the absence of radiation damping.