  Spatio-temporal evolution of a non-relativistic electrostatic waves in a cold
plasma has been studied in the wave-breaking regime using a 1D particle-in-cell
simulation. It is found that plasma gets heated after the wave-breaking but a
fraction of initial energy always remains with the remnant wave in the form of
BGK mode in warm plasma. An interesting finding of this work is that the
frequency of the resultant BGK wave is found be below electron plasma frequency
which decreases with increasing initial amplitude. Moreover, the acceleration
mechanism after the wave-breaking is also found to be different from the
previous work. In order to explain the results observed in the numerical
experiments, a simplified theoretical model is constructed which exhibits a
good agreement with the simulation. These investigations have direct relevance
in wake-field acceleration experiments.
