Infinitely many isolas of modulational instability for Stokes waves

This paper proves long-standing conjectures regarding the existence of infinitely many high-frequency modulational instability isolas" for a Stokes wave in arbitrary depth $ \mathtt{h} > 0 $, under longitudinal perturbations. We provide a complete characterization of the unstable spectral bands in the $L^2(\mathbb{R})$-spectrum of the water wave equations linearized around a Stokes wave of sufficiently small amplitude $ε$. The unstable spectrum is the union of isolated isolas" of elliptical shape, indexed by integers $\mathtt{p}\ge 2$, each with semiaxis of size $\mathrm{\mid }{\beta }_1^{(\mathtt{p})}(\mathtt{h})\mathrm{\mid }{\epsilon }^{\mathtt{p}}+O({\epsilon }^{\mathtt{p}+2})$. As first key achievement, we obtain an explicit formula for the coefficient ${\beta }_1^{(\mathtt{p})}(\mathtt{h})$ for any $\mathtt{p}\ge 2$, that remarkably depends solely on the maximal Taylor-Fourier coefficients of the Stokes wave. We provide simple expressions of the asymptotic expansion of such coefficients in the shallow-water limit $\mathtt{h}\to 0^{+}$, for any $\mathtt{p}\ge 2$. This allows to establish that the analytic function ${\beta }_1^{(\mathtt{p})}(\mathtt{h})$ is not zero for any $\mathtt{p}\ge 2$, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger.