  In this work we derive evolution equations for the nonlinear behavior of a
coasting beam under the influence of a resonator impedance. Using a
renormalization group approach we find a set of coupled nonlinear equations for
the beam density and the resonator voltage. Under certain conditions, these may
be analytically solved yielding solitary wave behavior, even in the presence of
significant dissipation in the resonator. We find long-lived perturbations,
i.e. droplets, which separate from the beam and decelerate towards a
quasi-equilibrium state, in good agreement with simulation results.
