Beam splitting can be performed by adiabatic crossing of a given one-dimensional non-linear resonance. This process is routinely used at the CERN PS to deliver the proton beam to the SPS fixed-target physics. To improve the efficiency of the intensity sharing between the various beamlets, a dipole kicker can be used to excite the beam during the resonance crossing process. This entails a double-resonance crossing phenomenon that will be described and discussed in detail in this paper.

In this paper, the nonlinear coupling resonance $2 Q_x -2 Q_y = 0$ is studied by means of a Hamiltonian model. The detailed analysis of its phase-space topology unveils the possible phenomena that can occur when crossing adiabatically such a resonance. These considerations are probed by means of numerical simulations carried out using a symplectic map and the results are presented and discussed in detail.

A new set of nonlinear beam manipulations have been recently proposed, with the goal of extending the transverse beam splitting that is routinely used at the CERN PS to deliver beam to the SPS for fixed-target physics. Using a simple Hamiltonian model, it has been shown how the transverse emittances can be shared by crossing a two-dimensional nonlinear resonance. Moreover, it has been shown how an AC-dipole can be used to split transversely the beam. In this paper, numerical simulations of these manipulations performed using a realistic model of the PS ring, including longitudinal motion, will be presented and discussed in detail.