Papers › Infinitely many isolas of modulational instability for Stokes waves
Infinitely many isolas of modulational instability for Stokes waves
Massimiliano Berti, Livia Corsi, Alberto Maspero, Paolo Ventura
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This paper proves long-standing conjectures regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth h > 0, under longitudinal perturbations. We provide a complete characterization of the unstable spectral bands in the L²(ℝ)-spectrum of the water wave equations linearized around a Stokes wave of sufficiently small amplitude ϵ. The unstable spectrum is the union of isolated ``isolas" of elliptical shape, indexed by integers p≥2, each with semiaxis of size |β₁⁽ᵖ⁾ (h)| ϵᵖ+ O(ϵᵖ⁺² ). As first key achievement, we obtain an explicit formula for the coefficient β₁⁽ᵖ⁾ (h) for any p ≥2, that remarkably depends solely on the maximal Taylor-Fourier coefficients of the Stokes wave. We provide simple expressions of the asymptotic expansion of such coefficients in the shallow-water limit h →0^+, for any p ≥2. This allows to establish that the analytic function β₁⁽ᵖ⁾(h) is not zero for any p ≥2, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger.
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