Frequency shifts of large-amplitude plasma waves

The time-asymptotic value of the frequency of a large-amplitude, electrostatic, electron plasma wave is studied over a wide range of amplitudes and phase velocities. When the amplitude is relatively small, the asymptotic frequency is less than the linear value, and the frequency shift increases with the square root of the amplitude. At larger amplitudes, the frequency shift increases more slowly, reaches a maximum, and then decreases. The asymptotic frequency finally becomes greater than the linear value