Vlasov solvers and simulation code analysis for mode coupling instabilities in both longitudinal and transverse planes

Two Vlasov solvers for the longitudinal and transverse planes are used to study the frequency shift of coherent oscillation modes and possible mode coupling instability for the two cases of a proton bunch interacting with either a constant inductive or a broad-band resonator impedance. In parallel to this approach, a new method to study the coherent frequency shift from the results of simulation codes is presented. Comparisons between the two methods are discussed, as well as simple analytical formulae (valid in the “long-bunch” regime), which clearly reveal how to mitigate these instabilities.