Thus, the kinetic equations (\[eqk1\],\[eqk2\]) will take the form $$\frac{\partial f}{\partial t}= \frac{4\pi ^2 e^2
}{m^2}\frac{\partial}{\partial v} \frac{W}{v}\frac{\partial
f}{\partial v}, \label{k1}$$ and $$\frac{\partial W}{\partial t}+\frac{v^2}{L_0} \frac{\partial
W}{\partial v}=\frac{\pi \omega_{pe}}{n}v^2W\frac{\partial
f}{\partial v}, \;\;\; \omega_{pe}=kv, \label{k2}$$ where the transport terms are omitted. Here, the inhomogeneity scale is also assumed to be constant and equal to $L_0$. It should be noted that this assumption is physically incorrect. The change in the spectrum of the Langmuir waves is due solely to the spatial movement of the waves with the group velocity. However, from a mathematical point of view, it is well justified as the group velocity of Langmuir waves is small ($3v_{Te}^2/v\ll v$) and effects connected with wave transport can be neglected.

Equations (\[k1\],\[k2\]) describe two physical effects: quasilinear relaxation (with characteristic time $\tau $) and the drift of Langmuir waves in velocity space (the characteristic time $\tau _2 = |L_0|/v$. Since $\tau _2\gg \tau$ the influence of plasma inhomogeneity can be considered as the evolution of the final stage of quasilinear relaxation. Two possible cases of plasma density change are considered: plasma density decreasing ($L_0<0$) with distance and plasma density increasing ($L_0>0$) with distance.

Plasma density decreasing with distance
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In this case 