It has been shown that the mode amplitude and frequency evolution in the early non-linear stage of a near-threshold bump-on-tail system can be reproduced by solving the linear dispersion relationship at each time step using the non-linearly modified distribution function at an earlier time. The dispersion relationship gives two solutions with departing frequencies almost immediately after the flattening of the distribution function starts to cancel out the drive. One can therefore attribute the early onset of the chirping directly to the modification of the underlying dispersion relationship. The existence of the two waves is because of the beam branch of the beam-plasma instabilities created by the perturbed distribution function. After the two chirping branches are formed, their frequencies are locked to the location of the peaks in the nonlinear distribution function, while the peaks are pushed forwards by beating itself. The transition from the beating-and-chirping scenario to chirping with hole-clump pair creation is found to be determined by the overlapping criterion of the two phase-space islands created by the two chirping branches.
This paper explores the early nonlinear stage of a near-threshold bump-on-tail instability, where the mode amplitude and frequency evolution can be reproduced by solving the linear dispersion relationship. The dispersion relationship gives two solutions with departing frequencies, leading to the early onset of frequency chirping. The transition from beating-and-chirping to chirping with hole-clump pair creation is determined by the overlapping criterion of the two phase-space islands.