gradient does not lead to the absorption of waves. This behavior results in different levels of plasma waves at two very close points with the opposite density-gradient sign.

Figure \[fig3\] demonstrates the existence of accelerated electrons with $v>v_0$. These electrons are accelerated by Langmuir waves in the regions with positive plasma-density gradient. Electrons with velocity larger than the initial beam velocity have been observed in laboratory plasma experiments. This effect was also considered from an analytical standpoint by ([@Ryutov69]) in application to laboratory plasmas.

Dynamics of electrons and accompanying Langmuir waves
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The processes of wave generation at the front and absorption at the back take place at every spatial point and therefore the structure can travel over large distances, being the source of plasma waves ([@Kontar98]; [@Melnik99]; [@Melnik00a]).

At time $t=5.0$s, electrons accompanied by Langmuir waves have passed over a large distance but the general physical picture remains the same (fig. \[fig4\]). Generally, electrons and Langmuir waves propagate as a BPS. At every spatial point electrons form a plateau at the electron distribution function and we have a high level of plasma waves. The electron cloud has a maximum of the electron density at $x=27d$. Plasma waves are also 