Bernstein–Greene–Kruskal modes

Bernstein–Greene–Kruskal modes (a.k.a. BGK modes) are nonlinear electrostatic waves that propagate in a collisionless plasma. They are nonlinear solutions to the Vlasov-Poisson system of equations 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 unmagnetized 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 (electrostatic solitary structures). and double layers in space plasmas, as well as in scattering experiments in the laboratory.