Phenomenological two-stream instability model in the nonlinear electron regime

We consider the e-p instability for long proton bunches in storage rings. We give a simplified treatment of the centroid model for the linear instability for both unbunched and bunched beams. We point out the very large resulting electron oscillation amplitudes which mean that the electrons very quickly become nonlinear. We then propose a phenomenological theory of the instability in the nonlinear electron regime and show that the resulting model gives a slower growth and different form of the proton oscillations than in the linear theory. We then propose a technique that might eliminate the long term secular growth of the proton oscillations