Multimode theory of electron hole transverse instability

We present Vlasov–Poisson three-dimensional linear stability analysis of an initially planar electron hole structure, solving for the distribution function by integration along unperturbed orbits. The non-sinusoidal potential perturbation shape (parallel to $B$ ) is expanded in eigenfunctions adiabatic Poisson operator, generalizing prior assumption a rigid shift equilibrium. show that shiftmode then modified second discrete mode plus integral over continuum wave-like modes. A rigorous treatment shows can be approximated effectively single satisfies external wave dispersion relation, thus making weighted sum three find numerically solution complex instability frequency, and corresponding amplitudes determining eigenmode. This multimode refines accuracy single-mode results, giving slightly higher growth rates at most parameters, as expected from extra freedom. Oscillating modes near boundaries have larger distortions which help explain particle-in-cell simulations observe up ${\sim }20$ % beyond thresholds, narrowing perturbation. At high magnetic field, predicts reduction already small rate.