  The Collective Beam-Beam interaction is studied in the framework of maps with
a ``kick-lattice'' model in the 4-D phase space of the transverse motion. A
novel approach to the classical method of averaging is used to derive an
approximate map which is equivalent to a flow within the averaging
approximation. The flow equation is a continuous-time Vlasov equation which we
call the averaged Vlasov equation, the new model of this paper. The power of
this approach is evidenced by the fact that the averaged Vlasov equation has
exact equilibria and the associated linearized equations have uncoupled
azimuthal Fourier modes. The equation for the Fourier modes leads to a Fredholm
integral equation of the third kind and the setting is ready-made for the
development of a weakly nonlinear theory to study the coupling of the pi and
sigma modes. The pi and sigma eigenmodes are calculated from the third kind
integral equation. These results are compared with the kick-lattice model using
our weighted macroparticle tracking code and a newly developed, density
tracking, parallel, Perron-Frobenius code.
