On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas

Quantum statistics and electron trapping have a decisive influence on the propagation characteristics of coherent stationary electrostatic waves. The description of these strictly nonlinear structures, which are of electron hole type and violate linear Vlasov theory due to the particle trapping at any excitation amplitude, is obtained by a correct reduction of the three-dimensional Fermi-Dirac distribution function to one dimension and by a proper incorporation of trapping. For small but finite amplitudes, the holes become of cnoidal wave type and the electron density is shown to be described by a ϕ(x)1/2 rather than a ϕ(x) expansion, where ϕ(x) is the electrostatic potential. The general coefficients are presented for a degenerate plasma as well as the quantum statistical analogue to these steady state coherent structures, including the shape of ϕ(x) and the nonlinear dispersion relation, which describes their phase velocity.