Integrable tops and non-commutative torus

M. Olshanetsky

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arXiv · nlin/0203003

Integrable tops and non-commutative torus

M. Olshanetsky

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Abstract

We consider the hydrodynamics of the ideal fluid on a 2-torus and its Moyal deformations. The both type of equations have the form of the Euler-Arnold tops. The Laplace operator plays the role of the inertia-tensor. It is known that 2-d hydrodynamics is non-integrable. After replacing of the Laplace operator by a distinguish pseudo-differential operator the deformed system becomes integrable. It is an infinite rank Hitchin system over an elliptic curve with transition functions from the group of the non-commutative torus. In the classical limit we obtain an integrable analog of the hydrodynamics on a torus with the inertia-tensor operator $\bar\partial^2$ instead of the conventional Laplace operator $\partial\bar\partial$.

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M. Olshanetsky. 2002-03-04. Integrable tops and non-commutative torus. https://arxiv.org/abs/nlin/0203003

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