Abstract: The Vlasov-Poisson model for a boundless collisionless hydrogen plasma in a self-consistent electric field remains a fundamental model in several branches of modern physics including particle physics, electrodynamics, and plasma physics. The popularity of this model stems from its simplicity, clarity, and remarkable effectiveness when describing complex processes in the microcosm. Among applications of the model, addressing the issue of controlled thermonuclear fusion is the most crucial. Under an electrostatic approximation, a linear stability for one-dimensional (1D) dynamical equilibria in the boundless collisionless hydrogen Vlasov–Poisson plasma assuming a stationary isotropic in space, but anisotropic in velocities electron distribution along with a stationary uniform over space background of resting ions is investigated in this paper. Based on the direct Lyapunov method, we rigorously demonstrate that such local thermodynamic equilibria exhibit an absolute linear instability under 1D perturbations. We establish a priori exponential lower estimate for the growth rates and describe initial data of small 1D perturbations that grow in time, providing, thereby, sufficient conditions for linear practical instability. Our study specifically points out that the classical Newcomb–Gardner–Rosenbluth stability sufficient condition is applicable only within a particular incomplete, unclosed class of the considered perturbations, indicating a formal nature of this condition. To support these findings, the analytical examples of regarded dynamic equilibria and superimposed on them small 1D perturbations which grow over time according to the derived estimate were constructed.

Cite: Gubarev Y.G. , Ding M.
Study of stability for a subclass of stationary solutions to the Vlasov-Poisson equations for hydrogen plasma in a one-dimensional setting
14th International Conference on Mathematical Modeling in Physical Sciences 20-23 Oct 2025