GALACTIC and GALACLIC: Two Vlasov solvers for the transverse and longitudinal planes

GALACTIC and GALACLIC, two Vlasov solvers for the study, in the transverse and longitudinal plane respectively, of single-bunch coherent oscillation modes, were recently developed starting from the Vlasov equation and using a decomposition on the low-intensity eigenvectors, as proposed by Laclare and Garnier. The first Vlasov solver was used for instance to shed light on the destabilising effect of resistive transverse dampers and the second helped understanding the details of the mode-coupling behind some longitudinal microwave instabilities. Both theories are reviewed in detail, highlighting in particular the similarities and peculiarities of the two approaches.