Nonlinear Landau damping

In the phase after the linear phase, Fig. 2 indicates that the total kinetic energy of the particles oscillates with time, with the saturation level larger than the initial kinetic energy of the particles, i.e., there is net energy drawn from the wave by the particles. This is the nonlinear Landau damping, which clearly demonstrates that net energy exchange between waves and particles can be nonzero on the long time scale without including any collisional effect.