Nonlinear dispersion of stationary waves in collisionless plasmas

TL;DR

This paper derives a nonlinear dispersion relation for stationary waves in collisionless plasmas, revealing how different particle distributions affect wave frequency shifts with amplitude.

Contribution

It introduces a non-differential form of the nonlinear dispersion relation using a single-particle Hamiltonian and generalizes the dielectric function for electrostatic oscillations.

Findings

Smooth distributions lead to a square-root dependence of frequency shift on amplitude.

Beam-like distributions produce different power laws or logarithmic nonlinearities.

Analytical expressions for trapped particle contributions are derived.

Abstract

A nonlinear dispersion of a general stationary wave in collisionless plasma is obtained in a non-differential form from a single-particle oscillation-center Hamiltonian. For electrostatic oscillations in nonmagnetized plasma, considered as a paradigmatic example, the linear dielectric function is generalized, and the trapped particle contribution to the wave frequency shift $\Delta\omega$ is found analytically as a function of the wave amplitude $a$. Smooth distributions yield $\Delta\omega\sim a^{1/2}$, as usual. However, beam-like distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation.