On the theory of the nonlinear Landau damping

TL;DR

This paper presents an exact solution to the collisionless Vlasov equation, analyzing nonlinear Langmuir wave behavior and demonstrating that finite amplitude waves generally do not experience damping, except in the linear approximation.

Contribution

It provides the first exact solution to the nonlinear Vlasov equation and explores the nonlinear behavior of Langmuir waves, including conditions for instability.

Findings

Finite amplitude waves are generally not subject to damping.

Damping occurs only in the linear approximation with very small wave amplitudes.

Under certain resonance conditions, waves become unstable.

Abstract

An exact solution of the collisionless time-dependent Vlasov equation is found for the first time. By means of this solution the behavior of the Langmuir waves in the nonlinear stage is considered. The analysis is restricted by the consideration of the first nonlinear approximation keeping the second power of the electric strength. It is shown that in general the waves with finite amplitudes are not subject to damping. Only in the linear approximation, when the wave amplitude is very small, are the waves experiencing damping. It is shown that with the definite resonance conditions imposed, the waves become unstable.