# On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure

## Abstract

**:**

## 1. Introduction

## 2. Lie-algebraic Approach to the Vector Fields on the Torus

**Proposition**

**1.**

**Remark**

**1.**

## 3. The Structure of the Group Orbit of the Chaplygin Hydrodynamical System

## 4. The Geometric Structure of the Chaplygin Type Hydrodynamical Systems

**Theorem**

**1.**

**Proof.**

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Das, A. Integrable Models; World Scientifc: Singapore, 1989. [Google Scholar]
- Drazin, P.G.; Johnson, R.S. Solitons: An Introduction; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Takhtajan, L.A.; Faddeev, L.D. Hamiltonian Approach in Soliton Theory; Springer: Berlin/Heidelberg, Germany, 1987. [Google Scholar]
- Brunelli, J.C.; Gürses, M.; Zheltukhin, K. On the integrability of a class of Monge–Ampére equations. Rev. Math. Phys.
**2001**, 13, 529–543. [Google Scholar] - Doubrov, B.; Ferapontov, E.V.; Kruglikov, B.; Novikov, V.S. On a Class of Integrable Systems of Monge-Ampere Type. J. Math. Phys.
**2017**, 58, 063508. [Google Scholar] [CrossRef] [Green Version] - Pressley, A.; Segal, G. Loop Groups; Clarendon Press: London, UK, 1986. [Google Scholar]
- Hentosh, O.E.; Prykarpatsky, Y.A.; Blackmore, D.; Prykarpatski, A.K. Lie-algebraic structure of Lax–ato integrable heavenly equations and the Lagrange-d’Alembert principle. J. Geom. Phys.
**2017**, 120, 208–227. [Google Scholar] [CrossRef] - Ovsienko, V. Bi-Hamilton nature of the equation u
_{tx}= u_{xy}u_{y}− u_{yy}u_{x}. arXiv**2008**, arXiv:0802.1818v1. [Google Scholar] - Ovsienko, V.; Roger, C. Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1. Commun. Math. Phys.
**2007**, 273, 357–378. [Google Scholar] [CrossRef] [Green Version] - Błaszak, M. Classical R-matrices on Poisson algebras and related dispersionless systems. Phys. Lett. A
**2002**, 297, 191–195. [Google Scholar] [CrossRef] - Blackmore, D.; Prykarpatsky, A.K.; Samoylenko, V.H. Nonlinear Dynamical Systems of Mathematical Physics; World Scientific Publisher: Hackensack, NJ, USA, 2011. [Google Scholar]
- Reyman, A.G.; Semenov-Tian-Shansky, M.A. Integrable Systems; The Computer Research Institute Publication: Moscow/Izhvek, Russia, 2003. [Google Scholar]
- Arik, M.; Neyzi, F.; Nutku, Y.; Olver, P.J.; Verosky, J. Multi-Hamiltonian Structure of the Born-Infeld Equation. J. Math. Phys.
**1988**, 30, 1338–1344. [Google Scholar] [CrossRef] [Green Version] - Olver, P.J.; Nutku, Y. Hamiltonian Structures for Systems of Hyperbolic Conservation Laws. J. Math. Phys.
**1988**, 29, 1610–1619. [Google Scholar] [CrossRef] [Green Version] - Stanyukovich, K.P. Unsteady Motion of Continuous Media; Pergamon: New York, NY, USA, 1960; p. 137. [Google Scholar]

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Prykarpatskyy, Y.
On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure. *Symmetry* **2020**, *12*, 697.
https://doi.org/10.3390/sym12050697

**AMA Style**

Prykarpatskyy Y.
On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure. *Symmetry*. 2020; 12(5):697.
https://doi.org/10.3390/sym12050697

**Chicago/Turabian Style**

Prykarpatskyy, Yarema.
2020. "On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure" *Symmetry* 12, no. 5: 697.
https://doi.org/10.3390/sym12050697