Benjamin–Feir Instability of Stokes Waves in Finite Depth
Massimiliano Berti, Alberto Maspero, P. Ventura
Abstract
Abstract Whitham 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 $$ {\mathtt h} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> is larger than a critical threshold $$\texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\approx 1.363 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mi>WB</mml:mi> </mml:mstyle> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>1.363</mml:mn> </mml:mrow> </mml:math> . In this paper, we completely describe, for any finite value of $$ \mathtt h >0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> is turned on. We prove, in particular, the existence of a unique depth $$ \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mi>WB</mml:mi> </mml:mstyle> </mml:msub> </mml:math> , which coincides with the one predicted by Whitham and Benjamin, such that, for any $$ 0< \mathtt h < \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>h</mml:mi> <mml:mo><</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mi>WB</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> , the eigenvalues close to zero are purely imaginary and, for any $$ \mathtt h > \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mi>WB</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> , a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As $$ {\mathtt h} \rightarrow \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}^{\, +} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mrow> <mml:mi>WB</mml:mi> </mml:mrow> </mml:mstyle> <mml:mrow> <mml:mspace/> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> 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 $$ \texttt{h}\rightarrow +\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> (deep water) and $$ \mu \rightarrow 0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (long waves) do not commute. In finite depth, the four eigenvalues have all the same size $$\mathcal {O}(\mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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 $$\texttt{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> is carefully tracked.