A numerical study of bunched beam transverse e-p instability based on the centroid model

In a recent theoretical study of the transverse electron-proton (e-p) instability, an asymptotic solution has been found for the equations describing the centroid motion of the traversing proton bunch and the stationary background electrons. It was shown that the combination of finite proton bunch length, non-uniform proton line density, and the single-pass e-p interaction cause the instability to evolve intricately in space and time even in the linear regime. This paper reports a numerical study of the e-p instability based on the same centroid equations. The purpose of the work is to compare the numerical solution with the analytic solution and to use the numerical approach to investigate the early development of the instability not covered by the asymptotic solution. In particular, the instability threshold and the initial growth of the instability are studied for various proton-beam conditions, fraction of charge neutralization, and initial perturbations