Rigorous Bloch-Floquet analysis of the Euler equations confirms the Benjamin-Feir instability via figure-eight eigenvalue splitting and yields exact stability regions for gravity-capillary Stokes waves.
Wilton.On ripples.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (6), 29(173), 688–700
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Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves
Rigorous Bloch-Floquet analysis of the Euler equations confirms the Benjamin-Feir instability via figure-eight eigenvalue splitting and yields exact stability regions for gravity-capillary Stokes waves.