The Cerenkov interaction of a weak electron beam with a plasma is studied numerically. Three parts of the nonlinear evolution of the convective instability are discussed. In the first part the understanding of the saturation amplitude of the initial wave and its harmonics is extended. In the second part it is shown that a quasi‐linear‐like cascade process is responsible for the simutaneous appearance of beam particles and waves which move slower than the original beam velocity. These modes destroy the trapped particle oscillations of the initial wave. The third part of the instability is the development of fast waves whose phase velocities are greater than the original velocity of the beam. It is shown that these waves arise due to a mode coupling mechanism.