Two-Dimensional Quasi-Linear Evolution of the Electron-Beam-Plasma Instability

We report on the quasi-linear evolution of the electron-beam-plasma instability in a two-dimensional system. The numerical solutions of the basic equations show that a two-dimensional system evolves in a different way as compared with a one-dimensional system. After saturation the wave energy monotonically decreases with time while the width of its spectral distribution in $k$ space narrows rapidly. In some cases, the system reaches a state which is modulationally unstable, and consequently cannot be described by the weak-turbulence theory.