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TMF, 2007, Volume 150, Number 3, Pages 355–370 (Mi tmf5984)  

This article is cited in 1 scientific paper (total in 1 paper)

Elliptic hydrodynamics and quadratic algebras of vector fields on a torus

M. A. Olshanetsky

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin–Feigin–Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus.

Keywords: Euler hydrodynamic equation, ideal fluid, quadratic Poisson algebra

DOI: https://doi.org/10.4213/tmf5984

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English version:
Theoretical and Mathematical Physics, 2007, 150:3, 301–314

Bibliographic databases:

Received: 08.06.2006

Citation: M. A. Olshanetsky, “Elliptic hydrodynamics and quadratic algebras of vector fields on a torus”, TMF, 150:3 (2007), 355–370; Theoret. and Math. Phys., 150:3 (2007), 301–314

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Levin A. Olshanetsky M. Zotov A., “Relativistic Classical Integrable Tops and Quantum R-Matrices”, J. High Energy Phys., 2014, no. 7, 012  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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