[2] BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases,[1][3][4][5] and are believed to be produced by the two-stream instability.
A harmonic hole equilibrium of the Vlasov-Poisson system, which is correctly described as a complete solution, i.e. inclusively its phase velocity, by the Schamel method,[16] shows that nonlinearity persists even in the small amplitude limit.
[17][18][19][20] Another indication that this claim is unfounded is that nonlinear single modes prove to be unconditionally marginal stable in current-carrying plasmas regardless of the drift velocity between electrons and ions.
[24] The solutions for the QBGK modes were put forth by Lange et al. in 1996,[25] with potential applications to quantum plasmas.
[26][27] Classical and quantum BGK modes as well as their appearance in charged particle beams in storage rings and circular accelerators have been summarized in.