Some criteria for the symmetry of stratified water waves

TL;DR

This paper establishes sufficient conditions for the symmetry of two-dimensional stably stratified steady periodic gravity water waves by leveraging elliptic structures and maximum principles to apply a moving planes argument.

Contribution

It introduces a novel method combining elliptic structure analysis and maximum principles to prove symmetry in stratified water waves.

Findings

Symmetry is guaranteed under certain size regimes.

Maximum principles hold for the governing equations in these regimes.

The method applies to waves with monotonic profiles between crests and troughs.

Abstract

This paper considers two-dimensional stably stratified steady periodic gravity water waves with surface profiles monotonic between crests and troughs. We provide sufficient conditions under which such waves are necessarily symmetric. This is done by first exploiting some elliptic structure in the governing equations to show that, in certain size regimes, a maximum principle holds. This then forms the basis for a method of moving planes argument.