Intermittency of space charge waves

Intermittency of the nonlinear waves in one-dimensional flow of charged particles is studied. A two-mode solution is obtained in the frame moving with the space charge wave. The second mode corresponds to modulation of the fundamental (first) mode. A resonance interaction of the two modes can lead to a sharp change in the phase trajectory parameters, which is manifested by random variation of the wave amplitude. Calculations are simplified by using a method based on the formal quantization of the classical solution. The corresponding Schrödinger equation is obtained, the solution of which formally coincides with that of the well-known Coulomb problem. The probability of transitions between levels with different azimuthal quantum numbers corresponding to different amplitudes of the classical solution is calculated. The probability distribution function W(ΔΦ), where ΔΦ is the phase length of a regular (laminar) part of the trajectory, is determined using the “azimuthal number-angle” uncertainty relation.