Breathers and Fermi-Pasta-Ulam-Tsingou recurrence for resonant three-wave interactions

Resonant three-wave interactions occur frequently for surface and internal waves. Quadratic nonlinearities constitute the dominant features, in sharp contrast with the cubic nonlinear Schrödinger model of a narrow-band wave packet. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. We study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulan-Tsingou recurrence (FPUT) phenomenon. A 'cascading mechanism' provides an analytical verification, as the fundamental and sideband modes reach roughly the same order of magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also obtained, similar to experimental observations of optical pulses governed by the nonlinear Schrödinger equation. Such energy spectra can also elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of 'effective energy' can elucidate the 'regular' and 'staggered' FPUT patterns. These results can enhance the understanding the dynamics of the upper ocean.