2025/01/02 by Massimiliano Berti, Berti, Massimiliano, Livia Corsi +6
Earth and Planetary Sciences · Mathematics · #Coastal and Marine Dynamics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing #math.AP
paper · pdf · doi:10.48550/arxiv.2501.01390
openalex publication_date 2025/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 \mathttp ≥ 2 for any value of the depth \mathtth > 0 such that an explicit analytic function β1^(\mathttp)(\mathtth) is not zero. In [3] it is proved that the map \mathtth ↦ β1^(\mathttp)(\mathtth) is not identically zero for any \mathttp ≥ 2 by showing that lim_\mathtth → 0+β1^(\mathttp)(\mathtth) = - ∞ . In this manuscript we compute the asymptotic expansion of β1^(\mathttp)(\mathtth) in the deep-water limit \mathtth → + ∞ -- it vanishes exponentially fast to zero -- for \mathttp=2, 3, 4.