Free-Surface Hydrodynamics in Conformal Variables: Are Equations of Free-Surface Hydrodynamics on Deep Water Integrable?

TL;DR

This paper explores the potential integrability of free-surface hydrodynamics equations on deep water, proposing that these equations are integrable when viewed as perturbations of a compressed fluid state, enabling exact solutions.

Contribution

It introduces a new argument supporting the integrability of free-surface hydrodynamics equations based on perturbations of a compressed fluid solution.

Findings

Equations have an exotic solution with flat surface and compression.

Integrability allows construction of exact solutions.

Supports the hypothesis of complete integrability.

Abstract

The hypothesis on complete integrability of equations describing the potential motion of incompressible ideal fluid with free surface in 2-D space in presence and absence of gravity was formulated by Dyachenko and Zakharov in 1994 [1]. Later on, the same authors found that these equations have indefinite number of additional motion constants [2] that was an argument in support of the integrability hypothesis. In this article we formulate another argument in favor of this conjecture. It is known [3] that the free-surface equations have an exotic solution that keeps the surface flat but describes the compression of the whole mass of fluid. In this article we show that the free-surface hydrodynamic is integrable if the motion can be treated as a finite amplitude perturbation of the compressed fluid solution. Integrability makes possible to construct an indefinite number of exact solutions of the Euler equations with free surface.