[en] A self-consistent theory of slowly evolving potential waves in non magnetized homogeneous collisionless plasma is presented. The theory includes: 1) description of the evolution of the particle distribution function at an arbitrary but slow, as compared to oscillations of trapped particles, change in the shape, phase velocity and energy of the wave; 2) corroboration of the fact that in the process of the slow evolution the waveform remains symmetrical relative to extreme points whose number is retained if the initial distribution function does not include singularities; 3) examination of the evolution of the shape and phase velocity of the wave in a first approximation of the found iteration procedure; 4) derivation and analysis of the fundamental evolution equation from which the criteria of stationarity and stability of the waves follow. On the basis of the theory a study has been made into the evolution of waves in bounded plasma with a fast beam whose initial velocity exceeds the maximum phase velocity of the waves. It is found that the waves lose stability in such a system. The optimum initial beam particle distribution function over velocities is found