Stability condition for the drive bunch in a collinear wakefield accelerator

The beam breakup instability of the drive bunch in the structure-based collinear wakefield accelerator is considered and a stabilizing method is proposed. The method includes using the specially designed beam focusing channel, applying the energy chirp along the electron bunch, and keeping energy chirp constant during the drive bunch deceleration. A stability condition is derived that defines the limit on the accelerating field for the witness bunch