Thus, we are confronted with the initial value problem of electron cloud propagation in a plasma with density fluctuations. The problem is nonlinear and is characterized by three different time scales. The fastest process in the system is the quasilinear relaxation, on the quasilinear timescale $\tau \approx
n/n'\omega_{pe}$. The second timescale is that of processes connected with plasma inhomogeneity. Thirdly, there is the timescale of an electron cloud propagation in a plasma that significantly exceeds all other timescales.

Quasilinear relaxation and plasma inhomogeneity
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The main interaction in the system is beam – wave interaction governed by the quasilinear terms on the right hand side of equations (\[eqk1\],\[eqk2\]) . It is well-known that the unstable electron distribution function (\[g\_0\]) leads to the generation of plasma waves. The result of quasilinear relaxation for an electron beam homogeneously distributed in space is a plateau of the electron distribution function ([@Ryutov70]) $$\begin{aligned}
 \displaystyle f(v,t\approx \tau) =\left\{\begin{array}{ll}
              \displaystyle \frac{n'}{v_0},&\mbox{$v<v_0$}\\
              0,&\mbox{$v>v_0$}
              \end{array}
\right. \label{f_relax}\end{aligned}$$ and the spectral energy density $$\begin{aligned}
 \displaystyle W(v,t\approx \tau) =
              \displaystyle \frac{mn'}{v_0\omega_{pe}}
\int_0^v\left(1-\frac{v_0}{n'}g_0(v)\right)dv,\;\;v<v_0
\label{w_relax}\end{aligned}$$ where $g_0(v)$ is the initial distribution function of the beam.

In the case of an inhomogeneous plasma we can also consider relaxation of a homogeneously distributed beam. 