Longitudinal and transverse mode coupling instability: Vlasov solvers and tracking codes

The study of collective effects in circular accelerators can be tackled by solving numerically the Vlasov equation or by using tracking codes. The two methods are obtained with different approaches: Vlasov solvers consider a continuous distribution function and describe the beam with coherent oscillation modes in frequency domain (ending up usually with an eigenvalue system to solve), while tracking codes use macroparticles and wakefields in time domain. In this paper we present two Vlasov solvers for the study of collective effects (from impedances/wakefields only) which evaluate the frequency shift of coherent oscillation modes and possible mode coupling instability in the single-bunch case for both longitudinal and transverse planes. In the longitudinal plane the Vlasov solver also takes into account the potential well distortion due to the wakefields under some conditions. In parallel to this theoretical approach, tracking codes, which include collective effects, have been used as benchmark. In particular, starting from their results, we also propose a new method to study the frequency shift of coherent modes and compare it with the output of the Vlasov solvers.