Free-Surface Hydrodynamics in Conformal Variables: Are Equations of Free-Surface Hydrodynamics on Deep Water Integrable?
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2016
- DOI:
- 10.48550/arXiv.1604.04778
- arXiv:
- arXiv:1604.04778
- Bibcode:
- 2016arXiv160404778Z
- Keywords:
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- Mathematical Physics
- E-Print:
- 8 pages, 7 references. The content of this article was reported on VIIIth International Lavrentyev Readings, September 6-11, 2015, Novosibirsk, Russia