Variational Instability for Irrotational Water Waves in Finite Depth

TL;DR

This paper demonstrates the variational instability of small-amplitude irrotational water waves in finite depth by reformulating the problem and analyzing the spectral properties of the second variation operator.

Contribution

It introduces a novel approach combining pseudo-differential reformulation and spectral analysis to establish instability results for water waves.

Findings

Proves variational instability for small-amplitude solutions

Reformulates water wave problem as a pseudo-differential Euler-Lagrange equation

Uses spectral analysis and Plotnikov transformation to demonstrate instability

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.