  We study the evolution of the self-modulation instability using bunches with
finite rise times. Using particle-in-cell simulations we show that unlike long
bunches with sharp rise times, there are large variations of the wake
amplitudes and wake phase velocity when bunches with finite rise times are
used. These results show that use of bunches with sharp rise times is important
to seed the self-modulation instability and to ensure stable acceleration
regimes.
