\[12,14,15,28\].

This paper is organized as follows. Section 1 mainly introduces hybrid analytic norms. In section 2, we will prove cyclotron damping at the linear level. We will state the new observation and sketch the proof of main theorem in section 3. Section 4 is dedicated to the deflection estimates of the particles trajectory. Section 5 is the key section, which states the phenomena of plasma echoes. We will control the error terms in section 6, and give the iteration in section 7.

Before stating our main theorem, we assume that the electric induction is zero, then the Maxwell’s equations reduce to the following forms $$\begin{aligned}
 &\nabla\cdot E(t,x)=\int_{\mathbb{R}^{3}}f(t,x,v)dv,\quad \nabla\times B(t,x)=0,\quad \partial_{t} B(t,x)=-\nabla\times E(t,x),\quad \nabla\cdot B(t,x)=0.
 \end{aligned}$$ Now based on the assumption that the electric induction is zero, we first give two results of the Vlasov equation with the electric field $ E(t,x)$ and the magnetic field $B(t,x) $ on both the linear and the nonlinear levels satisfying the conditions $ E= W(x)\ast\rho(t,x),\partial_{t}B=\nabla_{x}\times E,$ where $W(x)$ is a vector function and satisfies $|\widehat{W(k)}|\leq\frac{1}{1+|k|^{\gamma}}$.

Now we state our main result as follows.

Let $f^{0}:\mathbb{R}^{3}\rightarrow\mathbb{R}_{+}$ be an analytic velocity profile, and assume $W(x)=(W_{1}(x),W_{2}(x),0):\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ and $ W(x)$ is an odd function on $x_{3}$ satisfying 