On the linearity and stability of electrostatic structures based on the Schamel equation

TL;DR

This paper clarifies the linear behavior and stability of electrostatic structures in collisionless plasmas using the Schamel equation, linking nonlinear modes to microscopic plasma dynamics and confirming findings with simulations.

Contribution

It offers a new understanding of the linear limit in the Schamel equation and introduces a potential shift approach to address positivity issues, connecting theory with PIC simulation results.

Findings

Marginal stability of solitary electron holes.

Identification of a dominant shift eigenmode.

Confirmation of theoretical predictions with PIC simulations.

Abstract

This paper contributes in the first part to the correct understanding of the linear limit in the Schamel equation (S-equation) from the perspective of structure formation in collisionless plasmas. The corresponding modes near equilibrium turn out to be nonlinear modes of the underlying microscopic Vlasov-Poisson (VP) system for which particle trapping is responsible. A simple shift of the electrostatic potential to a new pedestal leads to non-negativity and thus mitigates the positivity problem of the S-equation. The stability of a solitary electron hole (bright soliton), based on both the S-equation and an earlier transverse but limited VP instability analysis, exhibits marginal stability and linear perturbations in the form of the asymmetric shift eigenmode of a solvable Schr\"odinger problem. This finding of a dominant shift mode perturbation also seems to have been observed in a numerical PIC simulation, thus providing some confirmation for both theories.