A different derivation of conservation laws for water waves

TL;DR

This paper introduces a new nonlocal formulation for water waves that systematically derives conservation laws for irrotational and vorticity flows, revealing potential additional nonlocal conservation laws.

Contribution

It presents a novel derivation method for water wave conservation laws that does not depend on Lie symmetries, extending to flows with constant vorticity.

Findings

Derivation of Olver's eight conservation laws for various flow types

Identification of potential additional nonlocal conservation laws

Extension of conservation law derivation to flows with vorticity

Abstract

We consider a new nonlocal formulation of the water-wave problem for a free surface with an irrotational flow based on the work of Ablowitz, Fokas, and Musslimani and presented in the recent work of Oliveras. The main focus of the short paper is to show how one can systematically derive Olver's eight conservation laws not only for an irrotational fluid, but also for constant vorticity (linear shear flow) without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the existence of additional, nonlocal, conservation laws for the water-wave problem.