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 $\epsilon$. The unstable spectrum is the union of isolated ``isolas" of elliptical shape, indexed by integers $\mathtt{p}\geq 2$, each with semiaxis of size $|\beta_1^{(\mathtt{p})} (\mathtt{h})| \epsilon^\mathtt{p}+ O(\epsilon^{\mathtt{p}+2} )$. As first key achievement, we obtain an explicit formula for the coefficient $\beta_1^{(\mathtt{p})} (\mathtt{h})$ for any $\mathtt{p} \geq 2$, 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 $\mathtt{p} \geq 2$. This allows to establish that the analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h})$ is not zero for any $\mathtt{p} \geq 2$, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger.