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Funktsional. Anal. i Prilozhen., 2003, Volume 37, Issue 1, Pages 38–54 (Mi faa135)  

This article is cited in 27 scientific papers (total in 27 papers)

Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type

M. V. Pavlov, S. P. Tsareva

a Krasnoyarsk State Pedagogical University named after V. P. Astaf'ev

Abstract: We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal hydrodynamic type systems satisfying natural semisimplicity and homogeneity conditions, we prove necessary and sufficient conditions under which the third structure is local or corresponds to a metric of constant curvature. The results are illustrated by some well-known physical examples of such systems.

Keywords: Egorov metric, Hamiltonian structure, hydrodynamic type systems, Riemann invariant, Whitham equations, Benney chain

DOI: https://doi.org/10.4213/faa135

Full text: PDF file (217 kB)
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English version:
Functional Analysis and Its Applications, 2003, 37:1, 32–45

Bibliographic databases:

UDC: 514.7+517.956.35
Received: 05.06.2002

Citation: M. V. Pavlov, S. P. Tsarev, “Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type”, Funktsional. Anal. i Prilozhen., 37:1 (2003), 38–54; Funct. Anal. Appl., 37:1 (2003), 32–45

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. V. M. Buchstaber, D. V. Leikin, M. V. Pavlov, “Egorov Hydrodynamic Chains, the Chazy Equation, and $SL(2,\mathbb{C})$”, Funct. Anal. Appl., 37:4 (2003), 251–262  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. M. V. Pavlov, “New integrable $(2+1)$-equations of hydrodynamic type”, Russian Math. Surveys, 58:2 (2003), 386–387  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Pavlov M.V., Popowicz Z., “Non-polynomial conservation law densities generated by the symmetry operators in some hydrodynamical models”, J. Phys. A, 36:31 (2003), 8463–8472  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Pavlov M.V., “Integrable hydrodynamic chains”, J. Math. Phys., 44:9 (2003), 4134–4156  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Ferapontov E.V., Pavlov M.V., “Hydrodynamic reductions of the heavenly equation”, Classical Quantum Gravity, 20:11 (2003), 2429–2441  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. M. V. Pavlov, “The Boussinesq equation and Miura type transformations”, J. Math. Sci., 136:6 (2006), 4478–4483  mathnet  crossref  mathscinet  zmath
    7. M. V. Pavlov, “Classifying Integrable Egoroff Hydrodynamic Chains”, Theoret. and Math. Phys., 138:1 (2004), 45–58  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Abenda S., Grava T., “Modulation of Camassa-Holm equation and reciprocal transformations”, Ann. Inst. Fourier (Grenoble), 55:6 (2005), 1803–1834  crossref  mathscinet  zmath  isi  scopus
    9. Pavlov M.V., “The Kupershmidt hydrodynamic chains and lattices”, Int. Math. Res. Not., 2006, 46987, 43 pp.  mathscinet  zmath  isi  elib
    10. M. V. Pavlov, “Integrability of the Egorov systems of hydrodynamic type”, Theoret. and Math. Phys., 150:2 (2007), 225–243  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Pavlov M.V., “Algebro-geometric approach in the theory of integrable hydrodynamic type systems”, Comm. Math. Phys., 272:2 (2007), 469–505  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    12. Odesskii A.V., Sokolov V.V., “Integrable pseudopotentials related to generalized hypergeometric functions”, Selecta Math. (N.S.), 16:1 (2010), 145–172  crossref  mathscinet  zmath  isi  scopus
    13. Gibbons J., Lorenzoni P., Raimondo A., “Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type”, J Geom Phys, 60:9 (2010), 1112–1126  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. El G.A., Kamchatnov A.M., Pavlov M.V., Zykov S.A., “Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions”, J Nonlinear Sci, 21:2 (2011), 151–191  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Morrison E.K., Strachan I.A.B., “Modular Frobenius Manifolds and their Invariant Flows”, Int Math Res Not, 2011, no. 17, 3957–3982  mathscinet  zmath  isi  elib
    16. Bialy M., Mironov A., “New Semi-Hamiltonian Hierarchy Related to Integrable Magnetic Flows on Surfaces”, Cent. Eur. J. Math., 10:5 (2012), 1596–1604  crossref  mathscinet  zmath  isi  scopus
    17. Romano S., “4-Dimensional Frobenius Manifolds and Painlevé' VI”, Math. Ann., 360:3-4 (2014), 715–751  crossref  mathscinet  zmath  isi  scopus
    18. Bialy M., Mironov A.E., “Integrable Geodesic Flows on 2-Torus: Formal Solutions and Variational Principle”, J. Geom. Phys., 87 (2015), 39–47  crossref  mathscinet  zmath  adsnasa  isi  scopus
    19. S. V. Agapov, “Ob integriruemom geodezicheskom potoke v magnitnom pole na dvumernom tore”, Sib. elektron. matem. izv., 12 (2015), 868–873  mathnet  crossref
    20. Pavlov M.V. Vitolo R.F., “on the Bi-Hamiltonian Geometry of Wdvv Equations”, Lett. Math. Phys., 105:8 (2015), 1135–1163  crossref  mathscinet  zmath  isi  elib  scopus
    21. Romano S., “Frobenius Structures on Double Hurwitz Spaces”, Int. Math. Res. Notices, 2015, no. 2, 538–577  crossref  mathscinet  zmath  isi  elib  scopus
    22. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    23. Pavlov M.V., Tsarev S.P., “On local description of two-dimensional geodesic flows with a polynomial first integral”, J. Phys. A-Math. Theor., 49:17 (2016), 175201  crossref  mathscinet  zmath  isi  elib  scopus
    24. Agapov S.V., Bialy M., Mironov A.E., “Integrable Magnetic Geodesic Flows on 2-Torus: New Examples via Quasi-Linear System of PDEs”, Commun. Math. Phys., 351:3 (2017), 993–1007  crossref  mathscinet  zmath  isi  scopus
    25. Bulchandani V.B., “On Classical Integrability of the Hydrodynamics of Quantum Integrable Systems”, J. Phys. A-Math. Theor., 50:43 (2017), 435203  crossref  mathscinet  zmath  isi  scopus
    26. Brini A., Carlet G., Romano S., Rossi P., “Rational Reductions of the 2D-Toda Hierarchy and Mirror Symmetry”, J. Eur. Math. Soc., 19:3 (2017), 835–880  crossref  mathscinet  zmath  isi  scopus
    27. Pavlov V M. Stoilov N.M., “The Wdvv Associativity Equations as a High-Frequency Limit”, J. Nonlinear Sci., 28:5 (2018), 1843–1864  crossref  mathscinet  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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