Different type modulation effect on decreasing the stocharticity threshould in beam-beam interaction

Comparative investigations of different types of modulation and their combained effect on decreasing the stochasticity threshold in colliding beam installations are considered. The interaction of a single particle with a field of an infinitely short colliding beam is studied in the one-dimensional approximation. The betatron phase modulation and external perturbation amplitude modulation are considered. It is shown that overlapping of nonlinear resonances for the lack of modulation is a cause of stochastic instability. The dependence of the threshold value of the critical current density on the modulation amplitude is given. It is also shown that the phase modulation leads to formation of side resonances near every main resonance, where each of the side resonances is splitting that facilitates overlapping of all the subsidiary resonances