Papers › On higher order isolas of unstable Stokes waves
On higher order isolas of unstable Stokes waves
Massimiliano Berti, Livia Corsi, Alberto Maspero, Paolo Ventura
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We overview the recent result [3, Theorem 1.1] about the high-frequency instability of Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of {\it isolas} parameterized by an integer p ≥2 for any value of the depth h > 0 such that an explicit analytic function β₁⁽ᵖ⁾(h) is not zero. In [3] it is proved that the map h ↦β₁⁽ᵖ⁾(h) is not identically zero for any p ≥2 by showing that lim_(h →0^+)β₁⁽ᵖ⁾(h) = - ∞. In this manuscript we compute the asymptotic expansion of β₁⁽ᵖ⁾(h) in the deep-water limit h →+ ∞ -- it vanishes exponentially fast to zero -- for p=2, $3$, $4$.
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