Relativistic Non-linear Theory of Plasma Oscillations†

Abstract

ABSTRACT Non-linear theory of plasma oscillations is extended into the threshold relativistic region where radiation damping can be neglected. A general characterization of the stationary oscillatory modes is performed in terms of a phase-space analysis with a comparison of the non-relativistic behaviour. Two procedures are offered for delineation of the amplitude-dependent frequency of the fundamental vibration. One of these is readily adopted to the zero-current limit, allowing a development already featured for the relativistic harmonic oscillator. Thus regimes requiring relativistic classical or quantum mechanical treatment are identified. In general, very high density plasmas require classical relativistic treatment, although tenuous astrophysical plasmas, where extremely large amplitudes of ion-electron motion can be accommodated, also may be included; the relativistic quantum analysis enters when the plasmon energy 𝒦ωp approaches the rest mass energy of the vibrating charge. A critique of recent disclosures on non-linear plasma oscillations is set forth and a novel method for solution of a related class of non-linear differential equations presented.