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The theory and simulation of coherent resonant coupling due to space charge in coasting or 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 coefficient, which enables us to resolve the issue of KV anomalies by relating them to negative energy modes. The second emphasis of this study is with reference to linacs and a detailed discussion of ''stability charts'' describing resonant regions where approach to equipartition may occur.

The combined effect of space charge and nonlinear resonance on beam loss and emittance was measured in a benchmarking experiment over a 1.2 s long flat bottom at 1.4 GeV kinetic energy in the presence of a single controllable octupole. By lowering the working point towards the resonance, a gradual transition from a loss-free core emittance blowup to a regime dominated by continuous loss was found. We compare the observation with 3D simulations based on a new analytical space charge model and obtain good agreement in the emittance blowup regime. Our explanation is in terms of the synchrotron oscillation, which causes a periodic tune modulation due to space charge, and leads to trapping and detrapping on the resonance islands. For working points very close to the resonance this induces a beam halo with large radius. The underlying dynamics is studied in detail, and it is claimed that the predicted halo in conjunction with a reduced dynamic aperture for the real machine lattice is the source of the loss observed in the experiment.

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