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On the Theory of Trapped Particle Instability. I. General Formulation and Analysis for Weak Coupling Limit

A general formulation is developed to investigate the stability of a collisionless, one-dimensional plasma which sustains a large-amplitude, monochromatic electron plasma wave. A nonlinear dispersion relation is derived which governs the stability of a small-amplitude test wave. It is shown that untrapped electrons can produce a significant modification to the Kruer-Dawson-Sudan dispersion relation which was derived by considering only the effect of trapped electrons on the test wave. As an application a weak-coupling limit is considered. It is shown that in this limit waves can grow in the distinct wavenumber regions given by \(|\varDelta k|{=}\sqrt{2n+1}k_{B}(n{=}0,1,2,\cdots)\), and | Δ k |≪ k B where Δ k is the difference of the wavenumbers of the test wave and the original large-amplitude wave and k B the bounce frequency divided by the phase velocity of the large-amplitude wave. Various characteristics of the growing waves are obtained.