On higher order isolas of unstable Stokes waves

TL;DR

This paper analyzes the high-frequency instability of Stokes waves, focusing on the asymptotic behavior of spectral isolas in deep water, providing explicit calculations for certain perturbation modes.

Contribution

It computes the asymptotic expansion of the spectral function ^{()}() in the deep-water limit for specific modes, extending previous results on instability.

Findings

Spectral isolas vanish exponentially fast as water depth increases.

Explicit asymptotic formulas are derived for modes p=2,3,4.

The instability map is non-trivial for all depths.

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 $ \mathtt{p} \geq 2 $ for any value of the depth $ \mathtt{h} > 0 $ such that an explicit analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not zero. In [3] it is proved that the map $ \mathtt{h} \mapsto \beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not identically zero for any $ \mathtt{p} \geq 2 $ by showing that $ \lim_{\mathtt{h} \to 0^+}\beta_1^{(\mathtt{p})}(\mathtt{h}) = - \infty $. In this manuscript we compute the asymptotic expansion of $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ in the deep-water limit $ \mathtt{h} \to + \infty $ -- it vanishes exponentially fast to zero -- for $\mathtt{p}=2$, $3$, $4$.