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Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 1, Pages 39–52 (Mi faa2890)  

This article is cited in 15 scientific papers (total in 15 papers)

The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets

O. I. Mokhovab

a M. V. Lomonosov Moscow State University
b Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: In this paper, the well-known Dubrovin–Novikov problem posed as long ago as in 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number $N$ of components and for any dimension $n$ by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.

Keywords: multidimensional Dubrovin–Novikov bracket, multidimensional Poisson bracket of hydrodynamic type, obstruction tensor, infinite-dimensional Lie algebra, compatible metrics, flat pencil of metrics, system of hydrodynamic type, compatible Poisson brackets

DOI: https://doi.org/10.4213/faa2890

Full text: PDF file (233 kB)
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English version:
Functional Analysis and Its Applications, 2008, 42:1, 33–44

Bibliographic databases:

UDC: 514.7+517.9
Received: 11.08.2006

Citation: O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funktsional. Anal. i Prilozhen., 42:1 (2008), 39–52; Funct. Anal. Appl., 42:1 (2008), 33–44

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Bogoyavlenskij O., Reynolds P., “Form-invariant Poisson brackets of hydrodynamic type with several spatial variables”, J. Math. Phys., 49:5 (2008), 053520, 15 pp.  crossref  mathscinet  zmath  adsnasa  isi
    2. Ferapontov E.V., Odesskii A.V., Stoilov N.M., “Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions”, J. Math. Phys., 52:7 (2011), 073505, 28 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Ferapontov E.V., Novikov V.S., Stoilov N.M., “Dispersive deformations of Hamiltonian systems of hydrodynamic type in $2+1$ dimensions”, Phys. D, 241:23-24 (2012), 2138–2144  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Maltsev A.Ya., “The Multi-Dimensional Hamiltonian Structures in the Whitham Method”, J. Math. Phys., 54:5 (2013), 053507  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Pavlov M.V., “Hamiltonian Formalism of Two-Dimensional Vlasov Kinetic Equation”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 470:2172 (2014), 20140343  crossref  mathscinet  zmath  adsnasa  isi
    6. Casati M., “On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type”, Commun. Math. Phys., 335:2 (2015), 851–894  crossref  mathscinet  zmath  adsnasa  isi
    7. Ferapontov E.V., Lorenzoni P., Savoldi A., “Hamiltonian Operators of Dubrovin-Novikov Type in 2D”, Lett. Math. Phys., 105:3 (2015), 341–377  crossref  mathscinet  zmath  adsnasa  isi
    8. Savoldi A., “Degenerate First-Order Hamiltonian Operators of Hydrodynamic Type in 2D”, J. Phys. A-Math. Theor., 48:26 (2015), 265202  crossref  mathscinet  zmath  isi  elib
    9. Della Vedova A. Lorenzoni P. Savoldi A., “Deformations of non-semisimple Poisson pencils of hydrodynamic type”, Nonlinearity, 29:9 (2016), 2715–2754  crossref  mathscinet  zmath  isi  scopus
    10. Maltsev A.Ya., “On the canonical forms of the multi-dimensional averaged Poisson brackets”, J. Math. Phys., 57:5 (2016), 053501  crossref  mathscinet  zmath  isi  elib  scopus
    11. Carlet G., Casati M., Shadrin S., “Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets”, J. Geom. Phys., 114 (2017), 404–419  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. Sergyeyev A., “New Integrable (3+1)-Dimensional Systems and Contact Geometry”, Lett. Math. Phys., 108:2 (2018), 359–376  crossref  mathscinet  zmath  isi
    14. Carlet G., Casati M., Shadrin S., “Normal Forms of Dispersive Scalar Poisson Brackets With Two Independent Variables”, Lett. Math. Phys., 108:10 (2018), 2229–2253  crossref  mathscinet  isi  scopus
    15. M. Casati, “Higher-order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type”, Theoret. and Math. Phys., 196:2 (2018), 1129–1149  mathnet  crossref  crossref  adsnasa  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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