Hosing Instability Suppression in Self-Modulated Plasma Wakefields

We show that the hosing instability can be suppressed after the saturation of the self-modulation instability of a long particle bunch if the plasma density perturbation is linear. We derive scalings for maximum bunch tilts and seeds for the self-modulation instability to ensure stable propagation beyond saturation of self-modulation instability. Numerical solutions of the reduced hosing equations and three-dimensional particle-in-cell simulations confirm our analytical findings. Our results may also apply when a train of particle bunches or laser pulses excites a linear wake.