The Three‐Wave Interaction—A Nondispersive Phenomenon

The one‐dimensional form of the three‐wave resonant interaction, for bounded envelopes, is solved exactly by an inverse scattering transform, which was originally suggested by Zakharov and Manakov. In addition to solving a third‐order inverse scattering problem, we show how to solve this more complicated third‐order three‐wave problem in terms of three simpler second‐order problems. The general result is that during the interaction, all solitons contained in the envelope of the wave with the middle group velocity will “split,” and be added to the original number of solitons contained in both the fast and slow envelopes. These results are then applied to the “decay” and “explosive” instabilities of plasmas, and necessary and sufficient conditions are found for exciting the explosive instability in the asymptotic limit of t →+∞.