bunched anisotropic equilibrium beams is presented. Our work confirms that analytical
results on coherent oscillations and instabilities of anisotropic KV (Kapchinskij-Vladimirskij)
distributions are a valid tool to interpret the findings from 2D and 3D self-consistent particle-
in-cell simulations for both KV and waterbag distributions. With reference to rings we discuss
space charge coherent tune shifts up to fourth order and introduce a coherent coupled mode …