Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy

TL;DR

This work studies the linear stability of Stokes waves in a deep, irrotational fluid using conformal variables and the Babenko equation. It proves rigorously that a zero eigenvalue bifurcation in the co-periodic stability problem occurs at every extremum of the energy $H$ as a function of steepness $s$, provided the wave speed $c$ does not extremize there, and it derives a Puiseux expansion and a normal form for the unstable eigenvalues. Numerically, it confirms that new unstable eigenvalues emerge in the direction of increasing steepness by computing generalized eigenfunctions and the coefficients of the normal form, at two energy extrema $s_1$ and $s_2$. The approach connects energy/momentum extrema to spectral bifurcations via a matrix-free, pseudo-differential formulation and paves the way for Bloch-Floquet analyses of modulational instabilities in Stokes waves, with potential extensions to broader spectral-band transformations.

Abstract

We address Euler's equations for irrotational gravity waves in an infinitely deep fluid rewritten in conformal variables. Stokes waves are traveling waves with the smooth periodic profile. In agreement with the previous numerical results, we give a rigorous proof that the zero eigenvalue bifurcation in the linearized equations of motion for co-periodic perturbations occurs at each extremal point of the energy function versus the steepness parameter, provided that the wave speed is not extremal at the same steepness. We derive the normal form for the unstable eigenvalues and, assisted with numerical approximation of its coefficients, we show that the new unstable eigenvalues emerge only in the direction of increasing steepness.

Figures (4)

• Figure 1: Schematic of oscillations of the Hamiltonian (green) and the speed (red) as the limiting Stokes wave is approached. The figure illustrates $H_{lim}-H$ and $c_{lim}-c$ as a function of $s_{lim}-s$, where $s_{lim}$, $H_{lim}$, and $c_{lim}$ represent the steepness, the Hamiltonian, and the speed of the limiting Stokes wave. The black circles mark the extreme points of the Hamiltonian, where the instability bifurcation occurs.
• Figure 2: The generalized eigenvectors $(v_1,w_1)$, $(v_2,w_2)$ and $(v_3,w_3)$ defined via equations \ref{['gen-Bab-eq']} with $a_1 = 1$, $a_2 = 0$, \ref{['gen-Bab-eq-2-red']} and \ref{['gen-Bab-eq-3']} (top to bottom) for the first two critical points of the Hamiltonian at $s_1 = 0.13660354990$ (left) and $s_2 = 0.14079654715$ (right).
• Figure 3: An example of numerical convergence of the iterative method for $(v_2,w_2)$ for the Stokes waves at $s_1 = 0.13660355$ (left) and $s_2 = 0.1407965471$ (right). The relative $L^2$ norm of the residual is shown versus the iteration number.
• Figure 4: Top left shows $\lambda^2(\varepsilon)$ obtained from numerical solution of the stability problem \ref{['spec-Bab-eq']} (red dots), and evaluating the expansion \ref{['halfseries']} (green line) with $\lambda_1^2 = 29.4871$ ($\mathcal{B} = 11.01822$); and top right shows a pair of real eigenvalues appearing from a collision of two imaginary eigenvalues at $s_{1}$ described in \ref{['halfseries']} with $c-c_1 = \varepsilon= -3.93\times10^{-5}$, $\varepsilon = -1.85\times10^{-5}$, $\varepsilon = -4.23\times 10^{-6}$ (gold, orchid and green triangles respectively), $\varepsilon=0$ (red circle), $\varepsilon = 4.17\times10^{-6}$, $\varepsilon = 1.78\times10^{-5}$ and $\varepsilon=3.75\times10^{-5}$ (green, orchid and gold diamonds respecively). Bottom row shows the same quantities at the second extremum at $s_2$ with $\mathcal{B} = 10.96232$.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Definition 1
• Theorem 1
• proof
• Remark 5
• Remark 6
• Remark 7