Nonlinear Wave Phenomena in Coasting Beams

In beams which are suf®ciently close to the linear stability limit, a variety of nonlinear wave phenomena are readily observed which can be used to diagnose aspects of the beam dynamics and the machine impedance. We have found that debunched beams in both the Fermilab Main Ring and the Tevatron are marginally stable to longitudinal oscillations and exhibit nonlinear three-wave coupling as well as nonlinear Landau damping and the formation of soliton-like perturbations. In addition, we have generated classical nonlinear echoes using twofrequency excitation in the Fermilab Accumulator. These phenomena can be used as diagnostic tools to determine diffusion rates in the beam as well as the longitudinal impedance. Moreover, aspects of these effects are likely involved in the approach to equilibrium as a longitudinal instability saturates under the in ̄uence of a driving impedance. We present experimental results and analytical models based on perturbation techniques. In addition, we describe particle simulations which model the fully nonlinear evolution of these phenomena.