Geodesic
Elliptic hydrodynamics and quadratic algebras of vector fields on a~torus
Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 3, pp. 355-370

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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.
Mots-clés : Euler hydrodynamic equation, quadratic Poisson algebra.
Keywords: ideal fluid 
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     author = {M. A. Olshanetsky},
     title = {Elliptic hydrodynamics and quadratic algebras of vector fields on a~torus},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {355--370},
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     volume = {150},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2007_150_3_a0/}
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M. A. Olshanetsky. Elliptic hydrodynamics and quadratic algebras of vector fields on a~torus. Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 3, pp. 355-370. http://geodesic.mathdoc.fr/item/TMF_2007_150_3_a0/