Stability of Bernstein-Greene-Kruskal Wave with Small Fraction of Trapped Electrons

For ω≫ωB = (k02e∅0/me)1/2, the normal modes are essentially oscillations of the background plasma and these can be made unstable due to coupling to the periodic motion of the trapped electrons with bound frequency ωb, where ω = nωb and n is an integer. However, the most unstable waves have frequency ω∼ωB, and in this regime, plasma oscillations with wave number k + k0 and k − k0, where k≪k0, are coupled; instability is associated with coupling due to resonant and nonresonant interaction with the trapped electrons. If the trapped electron distribution function is monotonically increasing (decreasing) in energy, then the lower (upper) frequency side band with wavenumber | k | − k0(| k |+k0) is dominant.