On higher order isolas of unstable Stokes waves

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 $\mathtt{p}\ge 2$ 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 $\mathtt{p}\ge 2$ by showing that ${\lim }_{\mathtt{h}\to 0^{+}}{\beta }_1^{(\mathtt{p})}(\mathtt{h})=-\mathrm{\infty }$. In this manuscript we compute the asymptotic expansion of ${\beta }_1^{(\mathtt{p})}(\mathtt{h})$ in the deep-water limit $\mathtt{h}\to +\mathrm{\infty }$ – it vanishes exponentially fast to zero – for $\mathtt{p}=2$, $3$, $4$.