Hydrodynamic modulation instability triggered by a two-wave system

The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in the form of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive media, such as hydrodynamics, optics, plasma, mechanical systems, electric transmission lines, and Bose-Einstein condensates, while its impact on applied sciences is steadily growing. It is well-known that the classical MI dynamics can be triggered when a pair of small-amplitude sidebands are excited within a particular frequency range around the main peak frequency. That is, a three-wave system, consisting of the carrier wave together with a pair of unstable sidebands, is usually adopted to initiate the wave focusing process in a numerical or laboratory experiment. Breather solutions of the nonlinear Schrödinger equation (NLSE) revealed that MI can generate much more complex localized structures, beyond the three-wave system initialization approach or by means of a continuous spectrum. In this work, we report an experimental study for deep-water surface gravity waves asserting that a MI process can be triggered by a single unstable sideband only, and thus, initialized from a two-wave process when including the contribution of the peak frequency. The experimental data are validated against fully nonlinear hydrodynamic numerical wave tank simulations and show very good agreement. The long-term evolution of such unstable wave trains shows a distinct shift in the recurrent Fermi-Pasta-Ulam-Tsingou focusing cycles, which are captured by the NLSE and fully nonlinear hydrodynamic simulations with some distinctions.