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Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations. (English. Russian original) Zbl 1106.37047

Funct. Anal. Appl. 40, No. 1, 11-23 (2006); translation from Funkts. Anal. Prilozh. 40, No. 1, 14-29 (2006).
The author solves the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics and establishes that this nontrivial special class of Hamiltonian operators is closely related to the associativity equations in two-dimensional topological quantum field and the theory of Frobenius manifolds. In particular, the author shows that any such Hamiltonian operator always defines integrable structural flows and generates integrable hierarchies of hydrodynamic type.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35Q58 Other completely integrable PDE (MSC2000)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
81V99 Applications of quantum theory to specific physical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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References:

[1] E. V. Ferapontov, ”Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funkts. Anal. Prilozhen., 25, No. 3, 37–49 (1991); English transl.: Functional Anal. Appl., 25, No. 3, 195–204 (19
[2] B. A. Dubrovin and S. P. Novikov, ”The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR, 270, No. 4, 781–785 (1983); English transl.: Soviet Math. Dokl., 27, 665–669 (19 · Zbl 0553.35011
[3] O. I. Mokhov and E. V. Ferapontov, ”Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature,” Usp. Mat. Nauk, 45, No. 3, 191–192 (1990); English transl.: Russian Math. Surveys, 45, No. 3, 218–219 (19 · Zbl 0712.35080
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[6] B. Dubrovin, ”Geometry of 2D topological field theories,” In: Integrable Systems and Quantum Groups, Lecture Notes in Math., Vol. 1620, Springer-Verlag, Berlin, 1996, pp. 120–348; http://arxiv.org/hep-th/9407018 ( · Zbl 0841.58065
[7] M. Kontsevich and Yu. I. Manin, ”Gromov-Witten classes, quantum cohomology, and enumerative geometry,” Comm. Math. Phys., 164, 525–562 (199 · Zbl 0853.14020
[8] O. I. Mokhov, ”Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations,” In: Topics in Topology and Mathematical Physics (S. P. Novikov, ed.), Amer. Math. Soc. Transl. Ser. 2, Vol. 170, Amer. Math. Soc., Providence, RI, 1995, pp. 121–151; http://arxiv.org/hep-th/9503076 · Zbl 0843.58048
[9] O. I. Mokhov, Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations [in Russian], IKI, Moscow-Izhevsk, 2004; English version: Reviews in Mathematics and Mathematical Physics, Vol. 11, Part 2, Harwood Academic Publishers, 2001.
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