Bernstein–Greene–Kruskal modes


Bernstein–Greene–Kruskal modes are nonlinear electrostatic waves that propagate in an unmagnetized, collisionless plasma. They are nonlinear solutions to the Vlasov–Poisson equation in plasma physics, and are named after physicists Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, who solved and published the exact solution for the one-dimensional case in 1957.
BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases, and are believed to be produced by the two-stream instability. They have been observed as electron phase space holes and double layers in space plasmas, as well as in scattering experiments in the laboratory.

Small-amplitude limit: Van Kampen modes

In the linear limit of BGK modes, the solutions reduce to what is known as Van Kampen modes, named after Nico van Kampen who derived the solutions in 1955. The phase mixing of Van Kampen modes gives rise to Landau damping.

Quantum BGK (QBGK) modes

BGK modes have been generalized to quantum mechanics, in which the solutions solve the quantum equivalent of the Vlasov–Poisson system known as the Wigner–Poisson system, with periodic boundary conditions. The solutions for the QBGK modes were put forth by Lange et al. in 1996, with potential applications to quantum plasmas.