Formation of rogue waves and modulational instability with zero-wavenumber gain

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Based on the integrable Bers-Kaup-Reiman system for the three-wave resonant interactions, we identify an RW-building instability of a novel type, viz., zero-wavenumber-gain MI, which contains zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by the MI in other multi-component systems.