A nonlinear dispersion relation (NDR) is a relation that assigns the obligatory phase velocity to an electrostatic wave structure propagating stationarily with in a collisionless plasma. This is done in accordance with the underlying equations, the Vlasov-Poisson system, and complements the formalism for determining the structure itself. The pseudo-potential method in the version of Schamel[1][2], termed S method, is thereby used, which is an alternative to the method described by Bernstein, Greene and Kruskal[3].
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 variety of electrostatic structures, including their phase velocities. These structures, also known under Bernstein–Greene–Kruskal modes or phase space electron and ion holes, or double layers, respectively, are ubiquitously found in collisionless plasmas such as in the Earth's magnetosphere, in fusion machines, or in the laboratory.
