ABSTRACT
This chapter discusses the results of numerical simulations of the self-consistent system, focusing on the single-wave model with N large. It also discusses the long-time evolution of the unstable wave, for the case of an initially cold beam and for an initially warm beam. The chapter considers the long-time evolution of the damping wave. Indeed the wave traps a significant fraction of particles, but the first trapped particles form a ‘macro-particle’ moving coherently in the resonance. The evolution with a linearly unstable wave has a rich history. Nonlinear superposition methods have been developed for finite-amplitude waves with non-overlapping resonance chains. A fairly detailed picture of the dynamics of several coupled modes was obtained by C. Lancellotti and J. Doming, who showed that initial data for the particle distribution function and the electric field evolve to asymptotic states with a finite number of field components.
