Stability and instability of small BGK waves

TL;DR

This work establishes a sharp, sign-based criterion for the linear stability of small BGK waves in the 1D Vlasov-Poisson system, tying stability to the sign of $\partial_e F(0,0)$, the slope of the energy distribution at zero energy. The authors introduce a dispersion operator $\mathbf{M}(\theta)$ that encapsulates the plasma response around a BGK background and develop a perturbative framework around homogeneous equilibria, leveraging a nonlinear pendulum reduction and infinite-dimensional modulation theory. They provide a comprehensive parameterization of BGK waves, a detailed decomposition of perturbations into BGK, Galilean, and dynamical components, and rigorous bounds for inner/outer/static contributions to the dispersion. The results yield linear instability for $\partial_e F(0,0)<0$ and linear stability (with a controlled decay) for $\partial_e F(0,0)>0$, with implications for the nonlinear behavior and the structure of the BGK manifold. The methodology combines action-angle coordinates, the Eddington transform, and a sophisticated operator-analytic treatment of the dielectric response to deliver precise spectral information near small BGK waves, enriching the understanding of BGK stability in weakly nonlinear regimes.

Abstract

The aim of this article is to prove that the linear stability or instability of small Bernstein-Green-Kruskal (BGK) waves is determined by the sign of the derivative of their energy distributions at $0$ energy.

Figures (5)

• Figure 1: Left: phase portrait of a homogeneous equilibrium. Right: phase portrait of a BGK wave. Dashed line shows the $0$-energy contour (separatrix) and the trapped region (negative energy) is in light grey.
• Figure 2: The function $K(k)$ (in blue), its equivalent given by (\ref{['ExpK2']}) (in red) and the function $E(k)$ (in black)
• Figure 3: Position of the pendulum on free trajectories as a function of the angle variable $\phi$ for different values of $a$. For $a$ close to $1$, the pendulum spends most of its time near $\pi$ (upright unstable position).
• Figure 4: Position of the pendulum on trapped trajectories as a function of the angle variable $\phi$ for different values of $a$. For $a$ close to $1$, the pendulum spends most of its time near $\pi$.
• Figure 5: Left: Curves of $F(E,\alpha)$ for various values of $\alpha$; Right: Plot of $\Re\{2\rho-\vert\xi\vert\}$ for the same equilibria.

Theorems & Definitions (83)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 1.4
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Definition 1.8
• Theorem 1.9
• Lemma 3.1