On the non-linear theory of electron beam and plasma interaction

A non-linear stationary solution for relativistic beam and homogeneous unbaunded plasma interaction is considered in hydrodynamics approach. The beam and the plasma are magnetized. That allows one to consider the motion to be one - dimensional. Non-linearity is connected with the wave effect on the electron beam motion. Plasma electron motion is thought to be linear. Non-linear wave equation has been obtained. The equation is of a finer character, than the known one. It's first integral has been found. This permits to determine the wave amplitude value excited by the beam in the plasma and, hence, the electromagnetic field energy as the current and the beam energy function. The utilization of the new method of the solution has allowed to enlarge the range of the theory application on the occasion of relativistic beam motion in the coordinate system moving with the phase wave velocity. It hasn't been considered earlier