Nonlinear dynamics of a driven mode near marginal stability

The nonlinear dynamics of a linearly unstable mode in a driven kinetic system is investigated to determine the saturated fields near the instability threshold. To leading order, this problem reduces to an integral equation with a temporally nonlocal cubic term. Its solution can exhibit self-similar behavior with a blowup in a finite time. When blowup occurs, the mode saturates due to plateau formation arising from particle trapping in the wave. Otherwise, the simplified equation gives a regular solution that leads to a saturation scaling reflecting the closeness to the instability threshold.