Nonlinear dispersion relation

A nonlinear dispersion relation (NDR) is a relation that assigns the correct phase velocity $v_{0}$ to a nonlinear wave structure. As an example of how diverse and intricate the underlying description can be, we deal with plane electrostatic wave structures $\phi (x-v_{0}t)$ which propagate with $v_{0}$ in a collisionless plasma. Such structures are ubiquitous, for example in the magnetosphere of the Earth, in fusion reactors or in the laboratory. Correct means that this must be done according to the governing equations, in this case the Vlasov-Poisson system, and the conditions prevailing in the plasma during the wave formation process. This means that special attention must be paid to the particle trapping processes acting on the resonant electrons and ions, which requires phase space analyses. Since the latter is stochastic and transient in nature, the entire dynamic trapping process eludes mathematical treatment, so that it can be adequately taken into account “only” in the asymptotic, quiet regime of wave generation, when the structure is close to equilibrium.

This is where the pseudo-potential method in the version of Schamel, also known as S-method, comes into play, which is an alternative to the method described by Bernstein, Greene and Kruskal. In the S method the Vlasov equations for the species involved are first solved and only in the second step Poisson's equation to ensure self-consistency. The S method is generally considered the preferred method because it is best suited to describing the immense diversity of electrostatic structures, including their phase velocities. These structures are also known under Bernstein–Greene–Kruskal modes or phase space electron and ion holes, or double layers, respectively.