Nonlinear dispersion of stationary waves in collisionless plasmas

A nonlinear dispersion of a general stationary wave in collisionless plasma is obtained in a nondifferential form expressed in terms of 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 Δω is found analytically as a function of the wave amplitude a. Smooth distributions yield Δω∼a1⊃/2, as usual. However, beamlike 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.