Dipole beam breakup electron cloud instability of a relativistic positron bunch with a smooth model linear density

We study the beam breakup instability of the dipole coherent oscillations of the relativistic positron bunch interacting with initially unperturbed electron cloud. We assume a special distribution of the linear density of positrons along the bunch which enables the calculations of analytic solutions of the linearized equations of motion of the cloud electrons. With these assumptions equations describing centroid beam breakup oscillations of the cloud and of the bunch can be solved taking into account effects of the cloud pinching and associated BNS damping. Although the BNS damping does not eliminate the electron cloud instability of coherent oscillations of the bunch, it changes the dependencies of the coherent oscillation amplitudes on the time from the quasi-exponential to the power function ones.