Novel solution of the system describing the resonant interaction of three waves

A novel solution is presented of the standard, integrable system of three coupled PDEs representing in 1+1 dimensions the three-wave resonant interaction phenomenon. In this new solution-a kind of “semi-dark single-soliton” solution-two of the three waves are localized (with a typical “solitonic” shape, vanishing asymptotically at large spatial distances), while the third has a “kink-like” shape, featuring a localized knee and being asymptotically finite. The time evolution of this solution displays quite a rich phenomenology: indeed, by focussing on its long-time behavior, one notes that, depending on the values of the parameters, this solution might display a phenomenon of pair annihilation or creation, namely each of its components might possess a pair of localized objects (soliton-like for the two asymptotically vanishing components, kink-like for the other) only in the remote past or only in the remote future, or it might instead possess a single such object both in the remote past and future but behaving overall as a boomeron or as a trappon, namely, in a reference frame moving with an appropriate constant velocity, coming in from one end in the remote past and boomeranging back there in the remote future, or being trapped to oscillate periodically around a finite position throughout the time evolution. And for special values of the parameters even more exotic phenomenologies emerge.