A Novel Method for Generating 3D Hydrodynamic Conservation Laws

TL;DR

This paper introduces a new nonlocal formulation for 3D water-wave problems that enables systematic derivation of conservation laws without Lie symmetries, revealing potential new nonlocal conservation laws.

Contribution

It develops a novel nonlocal formulation for 3D water-wave equations, allowing derivation of conservation laws independently of Lie symmetry methods.

Findings

Systematic derivation of Benjamin & Olver's twelve conservation laws.

Identification of potential additional nonlocal conservation laws.

Extension of previous work to three-dimensional water-wave problems.

Abstract

We extend the recent work of Oliveras arXiv:2008.00940 and Oliveras & Calatola-Young arXiv:2105.07580 to develop a new nonlocal formulation of the water-wave problem for a three-dimensional fluid with a two-dimensional free surface for an inviscid and irrotational fluid over a flat bathymetry. Using this new formulation, we show how one can systematically derive Benjamin & Olver's twelve conservation laws without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the potential existence of additional, nonlocal, conservation laws for the water-wave problem.