On the stability of weakly nonlinear short waves on finite-amplitude long gravity waves

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Abstract
The modulated nonlinear Schrödinger equation (Zhang & Melville 1990), describing the evolution of a weakly nonlinear short-gravity-wave train riding on a longer finite-amplitude gravity-wave train is used to study the stability of steady envelope solutions of the short-wave train. The formulation of the stability problem reduces to the solution of a pair of coupled equations for the disturbance amplitude and (relative) phase. Approximate analytical solutions and numerical solutions show that the conventional sideband (Benjamin-Feir) instability is just the first in a series of resonantly unstable regions which increase in number with increasing perturbation wavenumber. The first of these new instabilities is the result of a quintet resonance between four short waves and one long wave. Subsequent unstable regions correspond to sextet or higher-order resonances. The results presented here suggest that steady envelope solutions for unforced irrotational short waves on longer irrotational gravity waves may be unstable for a wide range of conditions.