Stabilisation of BGK modes by relativistic effects
  We investigate the acceleration of electrons via their interaction with electrostatic waves, driven by the relativistic Buneman instability, in a system dominated by counter-propagating proton beams. We observe the growth of these waves and their subsequent saturation via electron trapping for a range of proton beam velocities, from 0.15c to 0.9c. We can report a reduced stability of the electrostatic wave (ESW) with increasing non-relativistic beam velocities and an improved wave stability for increasing relativistic beam velocities, both in accordance with previous findings. At the highest beam speeds, we find the system to be stable again for a period of ~160 plasma periods. Furthermore we observe a, to our knowledge, previously unreported secondary electron acceleration mechanism at low beam speeds. We believe that it is the result of parametric couplings to produce high phase velocity ESW's which then trap electrons, accelerating them to higher energies. This allows electrons in our simulation study to achieve the injection energy required for Fermi acceleration, for beam speeds as low as 0.15c in unmagnetised plasma. 
N. J. Sircombe, M. E. Dieckmann, P. K. Shukla, T. D. Arber