([@Ryutov69]) $$\frac{\partial f}{\partial t}+v \frac{\partial f}{\partial x}=
\frac{4\pi ^2 e^2 }{m^2}\frac{\partial}{\partial v}
\frac{W}{v}\frac{\partial f}{\partial v}, \label{eqk1}$$ and $$\frac{\partial W}{\partial t}+\frac{\partial
\omega}{\partial k} \frac {\partial W}{\partial x}-\frac{\partial
\omega_{pe}}{\partial x } \frac{\partial W}{\partial k}=\frac{\pi
\omega_{pe}}{n}v^2W\frac{\partial f}{\partial v}, \;
\omega_{pe}=kv, \label{eqk2}$$ where $\partial \omega/\partial k=3v_{Te}^2/v$ is the group velocity of Langmuir waves, and $W(v,x,t)$ plays the same role for waves as the electron distribution function does for particles. The system (\[eqk1\],\[eqk2\]) describes the resonant interaction $\omega_{pe}=kv$ of electrons and Langmuir waves. On the right-hand side of equations (\[eqk1\],\[eqk2\]) I omit the spontaneous terms due to their small magnitude relative to the induced ones ([@Ryutov70]).

The presence of a local plasma frequency gradient leads to two physical effects on the kinetics of the Langmuir waves (\[eqk2\]). Firstly, the characteristic time of the beam-plasma interaction depends on the local density and therefore the resonance condition for the plasmons may itself change during the course of beam propagation. Secondly, the Langmuir wave propagating in the inhomogeneous plasma experiences a shift of wavenumber $\Delta k(x)$, due to the variation of the local refractive index. The second effect has been shown to have the main impact on Langmuir wave kinetics whereas the first effect can be neglected ([@Coste75]).

