Full Benjamin-Feir instability of capillary-gravity Stokes waves in finite depth

TL;DR

This work provides a rigorous, near-origin spectral description for small-amplitude 2D gravity-capillary Stokes waves in finite depth under long-wave perturbations. By combining Bloch-Floquet theory with a Kato-type perturbative framework and a non-perturbative block-decoupling step, the authors reduce the infinite-dimensional stability problem to a 4×4 Hamiltonian system whose eigenvalues yield a precise Benjamin-Feir spectrum. They identify an instability discriminant Δ_BF that, when positive, creates a figure-8 pattern of unstable eigenvalues near the origin, consistent with Modulational Instability in established unstable zones, and show how capillarity and depth shift the spectrum via the Whitham–Benjamin coefficients. The results connect formal modulational analyses of Djordjevic-Redekopp and Ablowitz-Segur with a rigorous spectral picture, providing a complete description of the near-origin spectrum and elucidating the stabilizing role of surface tension. The methodology combines Bloch theory, symplectic-normal form reductions, and delicate perturbative arguments to obtain a full BF spectrum for all κ ≥ 0 and h > 0.

Abstract

We study the two-dimensional gravity-capillary water waves equations for a fluid of finite depth $\mathtt{h}>0$ under the combined effects of gravity and surface tension $κ\geq 0$. We analyze the linear stability and instability of small-amplitude, $2π$-periodic Stokes wave solutions, under the effect of longitudinal long-wave perturbations. The corresponding linearized operator has periodic coefficients and a defective zero eigenvalue of multiplicity four. Using Bloch-Floquet theory, we investigate the associated family of periodic eigenvalue problems. For all surface tension values $κ\geq 0$ and depths $\mathtt{h} > 0$, we establish the complete splitting of the four eigenvalues near zero when both the wave amplitude and the Floquet parameter are small. Specifically, we rigorously prove that in the regions of unstable depth and capillarity identified formally by Djordjevic-Redekopp and Ablowitz-Segur in the 1970's, the spectrum of the linearized operator near the origin depicts a "figure 8" pattern.

Figures (1)

• Figure 1: Stability diagram for capillary–gravity Stokes wave. The capillarity $\kappa$ is on the horizontal axis and the depth $\mathtt{h}$ on the vertical axis. The light gray region is the unstable region of parameters $\mathcal{U}$ in \ref{['def:cU']}, while the white region is the stable region $\mathcal{S}$. Lines 1 and 5 (black color) are the zero set of the function $\mathsf{e}_{\mathrm{WB}}$ in \ref{['def:eWB']}. Line 2 (green color) is the zero set of the function $\mathsf{e}_{22}$ in \ref{['def:e22']}. Line 3 (red color) is the curve $(1+\kappa) \mathtt{c}_{\mathtt{h}}^4 - 3\kappa = 0$, i.e. the curve defining the set $\mathfrak{R}_2$, c.f. \ref{['def:fR']}. Line 4 (blue line) is the zero set of the function $\textup{D}_{\mathtt{h},\kappa}$ in \ref{['def:D']}. Line 6 (orange line) is the zero set of the function $\breve{\mathtt c}_{\mathtt{h},\kappa}$. Lines 3 and 4 are the singular lines of the function $\mathsf{e}_{\mathrm{WB}}$ on which the denominator vanishes. We also mark the curves $\mathfrak{R}_3,\mathfrak{R}_4,\ldots,\mathfrak{R}_{10}$ and $\mathfrak{R}_{10},\mathfrak{R}_{20},\ldots,\mathfrak{R}_{100}$, which are plotted as dotted lines and arranged in order from right to left.

Theorems & Definitions (55)

• Theorem 1.1: Benjamin--Feir unstable eigenvalues
• Theorem 2.1: Stokes waves
• Lemma 2.2
• Theorem 2.3: Complete Benjamin-Feir spectrum
• Remark 2.4
• Lemma 3.1
• Lemma 3.2
• Definition 3.3: Symplectic and reversible basis
• Remark 3.4: Parity structure of the reversible basis \ref{['basis is reversible']}
• Remark 3.5: Symplectic expansion using the basis in \ref{['basis is symplectic']}