---
author:
- 'Xixia Ma [^1]       \'
title: '**Cyclotron Damping along a Uniform Magnetic Field** '
---

**Abstract.** We prove cyclotron damping for the collisionless Vlasov-Maxwell equations on $\mathbb{T}_{x}^{3}\times\mathbb{R}_{v}^{3}$ under the assumptions that the electric induction is zero and $(\mathcal{\mathbf{PSC}})$ holds. It is a crucial step to solve the stability problem of the Vlasov-Maxwell equations. Our proof is based on a new dynamical system of the plasma particles, originating from Faraday Law of Electromagnetic induction and Lenz’s Law. On the basis of it, we use the improved Newton iteration scheme to show the damping mechanism.

[**1. Introduction** ]{}

In this paper, it is assumed that the plasma system is collisionless, nonrelativistic, and hot. Cyclotron damping describes the phenomenon that a plasma with a prescribed zero-order distribution function, imbedded in a uniform magnetic field, which is assumed to be perturbed by an electromagnetic wave propagating parallel to the field. A number of treatments of the problem of cyclotron damping have appeared in the literature\[18,20,24,25\], but there are few rigorous mathematical results except the recent results of Bedrossian and Wang \[7\] . The usual method that deals with this phenomenon is via the Vlasov equations. In this paper we study cyclotron damping 