Unstable BGK solitary waves and collisionless shocks

Consider a collisionless relativistic neutral plasma. We generalize the Penrose condition for linearized instability to the relativistic case. Then we consider a general one-dimensional equilibrium (a BGK wave) that is a collisionless shock or a solitary wave. The electric potential undergoes a transition from one constant to another as x runs from -infinity to +infinity. We prove that if one of these constants satisfies the relativistic version of the Penrose condition, then the BGK wave is nonlinearly unstable. We also prove that the periodic relativistic BGK waves of small amplitude are nonlinearly unstable.