quasilinear relaxation, electrons form a plateau in the electron distribution function and generate a high level of plasma waves. At time $t=0.1$s, the typical result of quasilinear relaxation is observed. The electron distribution function and the spectral energy density evolve in accordance with the gas-dynamic solution ([@Melnik95]; [@Melnik00a]) $$\begin{aligned}
 \displaystyle f(v,x,t) =\left\{\begin{array}{ll}
              \displaystyle \frac{n'}{v_0}\mbox{exp}\left(-\frac{(x-v_0t/2)^2}{d^2}\right),
              &\mbox{$v<v_0$}\\
              0,&\mbox{$v>v_0$}
              \end{array}
\right. \label{f_hom}\end{aligned}$$ $$\begin{aligned}
\label{w_hom} W(v,x,t)=\frac{mn'}{v_0\omega _{pe}}v^4
              \left[1-\frac{v}{v_0}\right]
              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cr
              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
              \times\mbox{exp}\left(-\frac{(x-v_0t/2)^2}{d^2}\right), \;\;v<v_0\end{aligned}$$ At this stage the influence of the plasma inhomogeneity is not observable.

The numerical solution of the kinetic equations and the gas-dynamic solution show that electrons propagate in a plasma accompanied by a high level of plasma waves. Since the plasma waves exist at a given point for some time, while the structure passes this point, the spectrum of the waves should change due to the wave movement. To understand the physics of the process we consider the evolution of the electron distribution function and the spectral energy density of Langmuir waves at a given point.

The electron distribution function and the spectral energy density of plasma waves
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At every spatial point we observe two physical processes. The first process is connected with the spatial movement of a BPS, as would 