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Funktsional. Anal. i Prilozhen., 2006, Volume 40, Issue 1, Pages 14–29 (Mi faa15)  

This article is cited in 11 scientific papers (total in 11 papers)

Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations). We show that each $N$-dimensional Frobenius manifold can locally be represented by a special flat $N$-dimensional submanifold with flat normal bundle in a $2N$-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.

DOI: https://doi.org/10.4213/faa15

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English version:
Functional Analysis and Its Applications, 2006, 40:1, 11–23

Bibliographic databases:

UDC: 517.9
Received: 10.05.2004

Citation: O. I. Mokhov, “Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations”, Funktsional. Anal. i Prilozhen., 40:1 (2006), 14–29; Funct. Anal. Appl., 40:1 (2006), 11–23

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. I. Mokhov, “Systems of integrals in involution and associativity equations”, Russian Math. Surveys, 61:3 (2006), 568–570  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. O. I. Mokhov, “Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds”, Theoret. and Math. Phys., 152:2 (2007), 1183–1190  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. O. I. Mokhov, “Duality in a special class of submanifolds and Frobenius manifolds”, Russian Math. Surveys, 63:2 (2008), 378–380  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234  mathnet  crossref  mathscinet  zmath  isi  elib
    5. Sergyeyev A., “Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations”, J. Phys. A, 42:40 (2009), 404017, 15 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    6. O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    7. Kath I., Nagy P.-A., “A Splitting Theorem for Higher Order Parallel Immersions”, Proc. Amer. Math. Soc., 140:8 (2012), 2873–2882  crossref  mathscinet  zmath  isi  elib  scopus
    8. Mikhail B. Sheftel, Devrim Yazici, “Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański”, SIGMA, 12 (2016), 091, 17 pp.  mathnet  crossref
    9. Sheftel M.B., Yazici D., Malykh A.A., “Recursion operators and bi-Hamiltonian structure of the general heavenly equation”, J. Geom. Phys., 116 (2017), 124–139  crossref  mathscinet  zmath  isi  scopus
    10. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. Prykarpatski A.K., “On the Solutions to the Witten-Dijkgraaf-Verlinde-Verlinde Associativity Equations and Their Algebraic Properties”, J. Geom. Phys., 134 (2018), 77–83  crossref  mathscinet  zmath  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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