Simulations of the strong-strong beam-beam interaction in hadron colliders

We develop weighted macro-particle tracking and the Perron-Frobenius operator technique for simulating the time evolution of two beams coupled via the beam-beam interaction. The π- and σ-modes, with and without a sextupole perturbation, are studied based on the Vlasov-Poisson system. Extending the standard averaging formalism to maps with a small collective force, we derive an approximation to the "kick-rotate" model of our simulation. This eliminates the delta function smoothing in the Vlasov equation, common in many theoretical approaches. Action densities are quasi-equilibria, consistent with simulation, and linearization leads to uncoupled Fourier modes and third-kind integral equations. Extensions to 2-d.o.f. and a flexible general purpose code are being developed