Bernstein–Greene–Kruskal modes

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.

Small-amplitude limit: Van Kampen modes ?

It is generally claimed that in the linear limit BGK modes (e.g. in the small amplitude approximation) reduce to what is known as Van Kampen modes, named after Nico van Kampen who derived the solutions in 1955.

This is wrong, however, since such a transition from a nonlinear to a linear mode does not take place even in the infinitesimal amplitude limit. 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, shows that nonlinearity persists even in the small amplitude limit. The area of trapped particles in the phase space never vanishes in this limit and there is no moment in which the distribution of trapped particles is transformed (or collapses) into a delta-function. 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. Landau's theory, as a linear wave theory, is obviously not applicable in case of coherent waves such as BGK modes valid even in the harmonic single wave limit. The advantage of the Schamel method over the BGK method, including the unlimited class of so-called undisclosed modes not covered by the BGK method, is discussed in and.