Effect of the long-term memory on the beam break-up instability of a single bunch in storage rings

We study modifications of the beam break-up instability of transverse coherent oscillations of a single bunch which occur in storage rings due to weak wakefields decaying longer than the revolution period of particles. The long-term part of the wake results in the eigenmode spectra of coherent oscillations. Both stable and unstable modes are found for coherent oscillations of a monochromatic bunch. The single turn wakefields result in the beam break-up coherent oscillations of the bunch. The found eigenmode spectrum does not contain a leading unstable mode. Despite the exponential increase in time of the eigenmodes, both self-consistent and the beam break-up parts of the coherent oscillations indicate similar and non-exponential time dependencies. The beam break-up behavior dominates, if the wake memory is weak.