In the our original paper the sign of the coefficient R 56 used in numeric calculations was wrong. It has to be changed to be consistent with the sign convention assumed in the design of a bunch compressor. The sign of R 56 depicted in Fig. 1 has to be changed from negative to positive. The sign of the chirp u related to the sign of the R 56 has to be changed as well: the chirp u in the first paragraph of Sec. VI should be positive, u 39:83 m ÿ1 . These changes do not affect the formulas of the paper but slightly modify the numeric results. The results of corrected calculations are shown in Figs. 1 and 2 given below, which should replace Figs. 3 and 4 in the paper, respectively.
A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave particle resonances just above the threshold for instability. The dominant nonlinearity stems from the dynamics of resonant particles that can be treated perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows the determination of conditions for a soft nonlinear regime, where the saturation level is proportional to the increment above threshold, or a hard nonlinear regime, characterized by explosive behavior, where the saturation level is independent of the closeness to threshold. In the hard regime, rapid oscillations typically arise that lead to large frequency shifts in a fully developed nonlinear stage. The universality of the approach suggests that the theory applies to many types of resonant particle driven instabilities, and several specific cases, viz. energetic particle driven Alfvén wave excitation, the fishbone oscillation, and a collective mode in particle accelerators, are discussed.
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