Nonlinear oscillations in a thin ring. II: Four-wave resonant interactions. Self-modulation (Q1384825)

The first part of the work [the author and \textit{A. I. Potapov}, ibid. 189-200 (1998)] has concluded that bending waves in ring, being the low-frequency modes of resonant triplet, are stable against small perturbations. Consequently, on the one hand, a bending wave should behave as a linear quasi-harmonic wave train. The first-order approximation analysis predicts that the triple mode interactions cannot play a primary role in the evaluation of bending waves. On the other hand, one may anticipate that the intense bending wavetrain can be subjected to the self-modulation during the long-time evolution. This means that the wave cannot be stable for a long time. The paper confirms such expectations by extending the first-order nonlinear analysis to include in the model the higher-order effects. Within the second-order approximation analysis, the amplitude modulation is examined for intense bending waves. As a result, the soliton-like envelopes can be formed from unstable bending wavetrains.