Nonlinear dispersion relation

Relation assigning the phase velocity

A nonlinear dispersion relation (NDR) is a dispersion relation that assigns the correct phase velocity 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 which propagate with 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, transient and rather filamentary 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.

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