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 isolas parameterized by an integer 𝚙≥2 for any value of the depth 𝚑>0 such that an explicit analytic function β^(𝚙)_1(𝚑) is not zero. In [3] it is proved that the map 𝚑↦β^(𝚙)_1(𝚑) is not identically zero for any 𝚙≥2. In this manuscript we compute the asymptotic expansion of β(𝚙)1(𝚑) in the deep-water limit 𝚑→+∞ – it vanishes exponentially fast to zero – for 𝚙=2, 3, 4.
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