  The long-time evolution of the two-stream instability of a cold ion beam
pulse propagating though the background plasma is investigated using a
large-scale one-dimensional electrostatic kinetic simulation. The three stages
of the instability are identified and investigated in detail. After the initial
linear growth and saturation by the electron trapping, a portion of the
initially trapped electrons becomes detrapped and moves ahead of the ion beam
pulse forming a {forerunner} electron beam, which causes a secondary two-stream
instability that preheats the upstream plasma electrons. Consequently, the
self-consistent nonlinear-driven turbulent state is set up at the head of the
ion beam pulse with the saturated plasma wave sustained by the influx of the
cold electrons from the upstream of the beam that lasts until the final stage
when the beam ions become trapped by the plasma wave. The beam ion trapping
leads to the nonlinear heating of the beam ions that eventually extinguishes
the instability.
