density fluctuations and $\alpha n_0$ is the amplitude of the density irregularities. The background plasma density is taken as a typical value for the starting frequencies of type III bursts $n_0=5\times
10^8$ cm$^{-3}$, corresponding to a local plasma frequency $f_p=\omega _{pe}/2\pi=200.73$MHz. As noted, small-intensity density fluctuation are considered, i.e. the local plasma frequency change due to the inhomogeneity is less than the thermal width of the plasma frequency (\[delta\_n\]). The value $\alpha$ is taken to be $10^{-3}$, which is considered to be a typical value for solar coronal observations ([@Cronyn72]; [@Smith79]). The spatial period of the plasma fluctuations $\Delta x=d/12$ is taken to be less than the initial size of the electron cloud. Thus, we have regions of size $\pi d/12\approx
0.26d$ with positive and negative density gradients. Recently, it has been shown that an electron beam can propagate in a homogeneous plasma as a BPS ([@Melnik99]; [@Melnik00]; [@Melnik00a]). Therefore, it is important to consider the dynamics of the electron beams at distances greatly exceeding the size of the electron cloud.

Initial evolution of the electron beam and formation of a BPS
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At the initial time $t=0$ we have an electron distribution function which is unstable. Due to fast 