Phase space dynamics after the breaking of a relativistic Langmuir wave in a thermal plasma

The relativistic Vlasov equation is integrated numerically in an Eulerian framework in order to investigate the phase space development of the wavebreak of a relativistic Langmuir wave in a thermal plasma. Relativistic kinematic and beam loading effects lead to a "differential retardation" in the time required by the electrons to evolve in phase space, the largest momentum electrons taking the longest time. This leads to the formation of a long lasting spike in momentum space at the wavebreak position that propagates with a velocity close to the speed of light and to an extremely steep density change in coordinate space.