Infinitely many isolas of modulational instability for Stokes waves

TL;DR

This paper proves the existence of infinitely many high-frequency modulational instability isolas for Stokes waves in water of arbitrary depth, providing explicit formulas and asymptotic analysis of their spectral properties.

Contribution

It offers a complete characterization of the unstable spectral bands and explicit formulas for the coefficients defining the isolas, advancing understanding of water wave stability.

Findings

Infinite number of instability isolas characterized

Explicit formulas for spectral coefficients derived

Asymptotic behavior in shallow water analyzed

Abstract

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.