H. M. Yin

Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined.

Nonlinear dynamical systems exhibiting periodic patterns and recurrence phenomena occur frequently in fluids, optics, and plasma. Mechanisms leading to such states from a tranquil background or continuous wave have been studied intensively. Modulation instability and the nonlinear Schrödinger model constitute illuminating examples.

Abstract Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers.