The problem of the nonlinear time evolution of a cold beam‐plasma system, for which weak turbulence theory is well known to be inapplicable, is examined under the restrictions to one‐dimensional electrostatic oscillations and for systems where the ratio of the beam density to the background plasma density is a small parameter. In this case, it may be shown that the electrostatic field undergoes rapid growth to a state of meta‐equilibrium, followed by a slower time development. The mechanism for the nonlinear saturation of this growth is the trapping of beam electrons in the wave troughs of the electrostatic field. The existence of a unique single wave nonlinear Bernstein‐Greene‐Kruskal stationary state is established for this system and its properties (e.g., its energy content, wavelength, phase velocity) are evaluated. This Bernstein‐Greene‐Kruskal state is apparently approached closely by the system in the course of its time evolution. The predictions of the theories are compared with computer calculations and are found to be in good agreement.