WKB approximation) when the length of a Langmuir wave is much smaller than the size of the plasma inhomogeneity ([@Vedenov67]; [@Ryutov69]) $$\label{WKB1}
  \lambda \ll L,$$ where $$\label{WKB2}
 L\equiv \left(\frac {1}{\omega _{pe}}\frac{\partial \omega _{pe}}{\partial
 x}\right)^{-1},$$ is the scale of ambient plasma density fluctuations, and $\omega
_{pe}$ is the local electron plasma frequency. The plasma inhomogeneity changes the dispersion properties of Langmuir waves and if the intensity of density fluctuations is small then the dispersion relation can be written $$\label{dispersion} \omega (k,x) =\omega_{pe}\left[1+\frac
12\frac{\Delta n}{n}+\frac{3k^2v_{Te}^2}{2\omega_{pe}^2}\right],$$ where $v_{Te}$ is the electron thermal velocity. The intensity of the density fluctuations should be small ([@Coste75]) $$\label{delta_n}
\frac{\Delta n}{n} < \frac{3k^2v_{Te}^2}{\omega_{pe}^2},$$ to ensure that the corresponding fluctuations of local plasma frequency are within the thermal width of plasma frequency. Thus, for the typical parameters of the corona plasma (plasma density $n=5\times 10^8$cm$^{-3}$ or plasma frequency $f_p=\omega
_{pe}/2\pi \approx 200.73$MHz), and assuming a beam velocity $v_0=10^{10}$cm s$^{-1}$, the density fluctuations are limited to $\Delta n/n<10^{-2}$.

Quasilinear equations
---------------------

In the case of weak turbulence theory (\[weakt\]), and under the conditions of the WKB approximation (\[WKB1\],\[delta\_n\]), the evolution of the electron distribution function $f(v,x,t)$ and the spectral energy density $W(v,x,t)$ are described by the system of kinetic equations 