Derives explicit closed-form instability index for the Benjamin-Feir spectrum of hydroelastic Stokes waves, producing a three-parameter stability diagram in depth, surface tension and bending rigidity that recovers known limits and shows resonance disappearance for b≥1/14 or κ≥1/2.
The disintegration of wave trains on deep water. Part 1. Theory
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abstract
We make rigorous spectral stability analysis for non-resonant capillary-gravity waves as well as resonant Wilton ripples of sufficiently small amplitude. Our analysis is based on a periodic Evans function approach, developed recently by the authors for Stokes waves. On top of our previous work, we add to the approach new framework ingredients, including a two-stage Weierstrass preparation manipulation for the Periodic Evans function associated to the wave and the definition of a stability function as an analytic function of the wave amplitude parameter. These new ingredients are keys for proving stability near non-resonant frequencies and defining index functions ruling both stability and instability near non-zero resonant frequencies. We also prove that unstable bubble spectra near non-zero resonant frequencies form, at the leading order, either an ellipse or a circle and provide a justification for Creedon, Deconinck, and Trichtchenko's formal asymptotic expansion for the Floquet exponent. For non-resonant capillary-gravity waves for the stability near the origin of the complex plane, our stability results agree with the prediction from formal multi-scale expansion. New are our stability results near non-zero resonant frequencies. As the effects of surface tension vanish, our result recovers that for gravity waves. Also new are our stability results for Wilton ripples of small amplitude near the origin as well as near non-zero resonant frequencies.
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2026 3verdicts
UNVERDICTED 3representative citing papers
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.
Surface tension stabilizes the modulational instability of large-amplitude gravity-capillary waves at smaller values than weakly nonlinear theory predicts, with nonmonotonic dependence on tension.
citing papers explorer
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Benjamin-Feir spectrum of hydroelastic Stokes waves
Derives explicit closed-form instability index for the Benjamin-Feir spectrum of hydroelastic Stokes waves, producing a three-parameter stability diagram in depth, surface tension and bending rigidity that recovers known limits and shows resonance disappearance for b≥1/14 or κ≥1/2.
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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.
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On the stability of large-amplitude gravity-capillary surface waves
Surface tension stabilizes the modulational instability of large-amplitude gravity-capillary waves at smaller values than weakly nonlinear theory predicts, with nonmonotonic dependence on tension.