On higher order isolas of unstable Stokes waves

Massimiliano Berti; Livia Corsi; Alberto Maspero; Paolo Ventura

doi:10.1007/s40574-025-00500-8

paper

On higher order isolas of unstable Stokes waves

arXiv:2501.01390 · doi:10.1007/s40574-025-00500-8

Abstract

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 $p \geq 2$ for any value of the depth $h > 0$ such that an explicit analytic function $β_{1}^{(p)} (h)$ is not zero. In [3] it is proved that the map $h \mapsto β_{1}^{(p)} (h)$ is not identically zero for any $p \geq 2$ by showing that $lim_{h \to 0^{+}} β_{1}^{(p)} (h) = - \infty$ . In this manuscript we compute the asymptotic expansion of $β_{1}^{(p)} (h)$ in the deep-water limit $h \to + \infty$ -- it vanishes exponentially fast to zero -- for $p = 2$ , $3$ , $4$ .

Supported by PRIN 2020 (2020XB3EFL001), PRIN 2022 (2022HSSYPN), ERC Stg 2021 HamDyWWa (101039762), ERC Cog 2023 GUnDHam (101124921). arXiv admin note: substantial text overlap with arXiv:2405.05854