Benjamin–Feir Instability of Stokes Waves in Finite Depth

Abstract

AbstractWhitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth $$ {\mathtt h} $$ h is larger than a critical threshold $$\texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\approx 1.363 $$ h WB ≈ 1.363 . In this paper, we completely describe, for any finite value of $$ \mathtt h >0 $$ h > 0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent $$\mu $$ μ is turned on. We prove, in particular, the existence of a unique depth $$ \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ h WB , which coincides with the one predicted by Whitham and Benjamin, such that, for any $$ 0< \mathtt h < \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ 0 < h < h WB , the eigenvalues close to zero are purely imaginary and, for any $$ \mathtt h > \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ h > h WB , a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As $$ {\mathtt h} \rightarrow \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}^{\, +} $$ h → h WB + the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits $$ \texttt{h}\rightarrow +\infty $$ h → + ∞ (deep water) and $$ \mu \rightarrow 0 $$ μ → 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size $$\mathcal {O}(\mu )$$ O ( μ ) , unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth $$\texttt{h}$$ h is carefully tracked.