Variational Instability for Irrotational Water Waves in Finite Depth

TL;DR

This work addresses the variational stability of small-amplitude, periodic irrotational gravity waves in finite depth. By recasting the traveling-wave problem as a one-dimensional pseudo-differential Euler–Lagrange equation with a scalar constraint and applying a Plotnikov-type transform, the authors reduce the second variation to a diagonal form governed by a Plotnikov potential, then use spectral perturbation theory to track eigenvalues along the primary bifurcation branch. They prove that the trivial branch is variationally stable up to the first bifurcation, while the nontrivial branch becomes variationally unstable for small amplitudes; explicitly, the eigenvalue splitting near the first bifurcation is negative, signaling instability. The results highlight a finite-depth spectral gap as a stabilizing feature for the trivial state but confirm instability once nontrivial waves bifurcate, offering a rigorous foundation for future exploration of larger amplitudes and vorticity effects.

Abstract

We prove variational instability for small-amplitude solutions to the periodic irrotational gravity water wave problem in finite depth. Our results are based on a reformation of the water wave problem as a pseudo-differential Euler-Lagrange equation together with the local existence theory of small-amplitude waves. We use a perturbative spectral analysis of the second-variation operator in combination with a Plotnikov transformation to show instability for non-trivial solutions.

Figures (4)

• Figure 1: Schematic bifurcation diagram of solutions to \ref{['equ']}: the trivial solution branch $\mathcal{K}_{\rm triv}$ (blue line) is stable until the first bifurcation point $(h,\mu^*_1)$, where it becomes unstable (red line). The non-trivial solution branch $\mathcal{K}_1$ (red curve) is unstable for small amplitudes.
• Figure 2: Stability regions of the trivial solution $w=0$ to \ref{['equ']} in dependence of $h$ and $\mu$: If we only consider variations with fixed conformal mean depth, parameter values in the deep blue regions guarantee stability, while, if we also take into account variations in the conformal mean depth, the stronger stability condition of the light blue regions applies.
• Figure 3: The symbol of the Fourier multiplier $\mathcal{C}_{D}\partial_x$ in the finite-depth problem is given by $n\coth(Dn)$, depicted in red with $D=1$, while the symbol of the Fourier multiplier $\mathcal{C}\partial_x$ appearing in the infinite depth problem is given by $|x|$, depicted in black. The spectrum of $\mathcal{C}_{D}\partial_x$ thus has a spectral Gap of width $D^{-1}$, allowing for stability of the trivial solution, while any generic perturbation constant non-negative perturbation of the operator $\mathcal{C}\partial_x$ leads to negative eigenvalues and, therefore, instability of the constant solution.
• Figure 4: Contour of integration $\Gamma$ appearing in the proof of Lemma \ref{['Imparts']}.

Theorems & Definitions (18)

• Theorem 1.1
• Definition 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.4
• Remark 4.1
• Theorem 4.2
• Remark 4.3
• Remark 4.4