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We study the evolution of the self-modulation instability using bunches withfinite rise times. Using particle-in-cell simulations we show that unlike longbunches with sharp rise times, there are large variations of the wakeamplitudes and wake phase velocity when bunches with finite rise times areused. These results show that use of bunches with sharp rise times is importantto seed the self-modulation instability and to ensure stable accelerationregimes.

Self-modulation instability of ultra-relativistic particle bunches with finite rise times