Instability of large solitary water waves

TL;DR

This paper investigates the linear instability of 2D irrotational solitary water waves, revealing that certain large amplitude waves are inherently unstable, especially those approaching the highest wave with a 120-degree crest angle.

Contribution

It introduces a novel analysis linking linear instability to nonlocal dispersion operators and provides new insights into the instability of large amplitude solitary waves beyond small amplitude models.

Findings

Solitary waves higher than the maximal energy wave are linearly unstable.

Unstable waves can approach the highest wave with a 120-degree crest.

Large amplitude unstable waves cannot be captured by small amplitude models.

Abstract

We consider the linearized instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are not obtained at the highest wave, which has a 120 degree angle at the crest. Under the assumption of non-existence of secondary bifurcation which is confirmed numerically, we prove linear instability of solitary waves which are higher than the wave of maximal energy and lower than the wave of maximal travel speed. It is also shown that there exist unstable solitary waves approaching the highest wave. The unstable waves are of large amplitude and therefore this type of instability can not be captured by the approximate models derived under small amplitude assumptions. For the proof, we introduce a family of nonlocal dispersion operators to relate the linear instability problem with the elliptic nature of solitary waves. A continuity argument with a moving kernel formula is used to study these dispersion operators to yield the instability criterion.