On the existence, linearity and stability of electrostatic hole structures in the Vlasov–Poisson plasma from the perspective of its three velocity-separated evolution equations

An overview of nonlinear electrostatic structures in a collisionless plasma is given, as described by its three Schamel-type evolution equations. Separated in the phase velocity, these equations are related to the three acoustic modes of a two-component plasma, namely ion acoustic, the slow electron acoustic, and the slow ion acoustic mode. In their derivation, a novel coupling method is used that combines the propagation part with the structural part of the coherent wave pattern, with the focus on the exact reproduction of the kinetic equilibrium structures of the Vlasov–Poisson (VP) system. This is where the two central elements of Schamel’s equilibrium theory come into play, the nonlinear dispersion relation and the pseudo-potential. Various aspects such as existence, linearity, particle trapping scenario, non-negativity and stability are investigated and the corresponding fundamentals are conveyed. These include the correct understanding of the linear limit as distinct from the linear Vlasov limit and the alleviation of the positivity problem associated with the square root nonlinearity $$\sqrt{\phi }\partial _x\phi$$ by introducing appropriate pedestals for the electrostatic potential $$\phi (x,t)$$ . A general proof for the existence of solitary ion hole solutions over the entire temperature range is presented: $$0<\theta =\frac{T_e}{T_i} < \infty$$ , which corrects and extends the more restrictive condition $$\theta \le 3.5$$ used in the literature. Ion holes can therefore also exist for hotter electrons. The stability of a solitary electron hole, based on the S-equation, which focuses on a specific macroscopic structural behavior beyond kinetics, and a previous transverse but limited VP instability analysis, exhibits marginal longitudinal stability. The associated linear perturbations are in the form of the asymmetric shift eigenmode of a solvable Schrödinger problem. This finding of the possible dominance of the shift mode perturbation provides a new hint for the anticipated general kinetic proof of marginal stability and transverse instability of electrostatic structures under these conditions including undisclosed potentials.