$L_0$ is negative. After the time of quasilinear relaxation, a plateau is established in the electron distribution function and a high level of Langmuir waves is generated. Since the quasilinear processes are fast we have a plateau at every moment of time $$\begin{aligned}
 \displaystyle f(v,t\approx \tau) =\left\{\begin{array}{ll}
              \displaystyle \frac{n'}{v_0},&\mbox{$v<v_0$}\\
              0,&\mbox{$v>v_0$}
              \end{array}
\right. \label{relax2}\end{aligned}$$ The wave spectrum is changing with time, and from the fact that we have a plateau at every moment equation (\[k2\]) can be reduced to $$\frac{\partial W}{\partial t}-\frac {v^2} {|L_0|} \frac{\partial
W}{\partial v}= 0 \label{W}$$ The role of initial wave distribution is played by the spectral energy density generated during the relaxation stage (\[w\_relax\]). Integrating equation (\[W\]) we obtain the solution for $t\gg \tau$ $$\begin{aligned}
\label{w1} W(v,t)=\frac{m}{\omega _{pe}}(1/v-t/|L_0|)^{-3}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cr
\;\;\;
\times\int_0^{1/(1/v-t/|L_0|)}\left[\frac{n'}{v_0}-g_0(v)\right]dv,\;\;
              v<u(t)\end{aligned}$$ where $$\label{u}
  u(t)=\frac{v_0}{1+v_0t/|L_0|}$$ is the maximum velocity of the Langmuir waves. Note, that the electron distribution function is constant and presents a plateau (\[relax2\]).

![The electron distribution function $f(v,t)$ and the spectral energy density of Langmuir waves $W(v,t)$ at various times, for the case where the plasma density decreases with distance, $L_0=-5\times 10^9$cm. Numerical solution of kinetic equations (\[k1\],\[k2\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig1"}](ms1121f1.eps){width="90mm"}

The numerical solution of equations (\[k1\],\[k2\]) with the initial electron distribution function 