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TMF, 2010, Volume 163, Number 2, Pages 179–221 (Mi tmf6496)  

This article is cited in 13 scientific papers (total in 13 papers)

Integrable $(2+1)$-dimensional systems of hydrodynamic type

A. V. Odesskiiab, V. V. Sokolova

a Landau Institute for Theoretical Physics, RAS, Moscow, Russia
b Brock University, St. Catharines, Ontario, Canada

Abstract: We describe the results that have so far been obtained in the classification problem for integrable $(2+1)$-dimensional systems of hydrodynamic type. The Gibbons–Tsarev (GT) systems are most fundamental here. A whole class of integrable $(2+1)$-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus $g=0$ and $g=1$ and also a new GT system corresponding to algebraic curves of genus $g=2$. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial”.

Keywords: dispersionless integrable system, hydrodynamic reduction, Gibbons–Tsarev system

DOI: https://doi.org/10.4213/tmf6496

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English version:
Theoretical and Mathematical Physics, 2010, 163:2, 549–586

Bibliographic databases:

Received: 09.12.2009

Citation: A. V. Odesskii, V. V. Sokolov, “Integrable $(2+1)$-dimensional systems of hydrodynamic type”, TMF, 163:2 (2010), 179–221; Theoret. and Math. Phys., 163:2 (2010), 549–586

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Odesskii A.V., Sokolov V.V., “Classification of integrable hydrodynamic chains”, J. Phys. A, 43:43 (2010), 434027, 15 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Odesskii A.V. Sokolov V.V., “Non-Homogeneous Systems of Hydrodynamic Type Possessing Lax Representations”, Commun. Math. Phys., 324:1 (2013), 47–62  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Habibullin I., “Characteristic Lie Rings, Finitely-Generated Modules and Integrability Conditions for (2+1)-Dimensional Lattices”, Phys. Scr., 87:6 (2013), 065005  crossref  zmath  adsnasa  isi  elib  scopus
    4. Takashi Takebe, “Dispersionless BKP Hierarchy and Quadrant Löwner Equation”, SIGMA, 10 (2014), 023, 13 pp.  mathnet  crossref  mathscinet
    5. Ferapontov E.V. Kruglikov B.S., “Dispersionless Integrable Systems in 3D and Einstein-Weyl Geometry”, J. Differ. Geom., 97:2 (2014), 215–254  crossref  mathscinet  zmath  isi  scopus
    6. Ferapontov E.V. Moss J., “Characteristic Integrals in 3D and Linear Degeneracy”, J. Nonlinear Math. Phys., 21:2 (2014), 214–224  crossref  mathscinet  isi  scopus
    7. Ferapontov E.V. Moss J., “Linearly Degenerate Partial Differential Equations and Quadratic Line Complexes”, Commun. Anal. Geom., 23:1 (2015), 91–127  crossref  mathscinet  zmath  isi  scopus
    8. A. V. Odesskii, “Integrable structures of dispersionless systems and differential geometry”, Theoret. and Math. Phys., 191:2 (2017), 692–709  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.  mathnet  crossref
    10. Morozov O.I. Pavlov M.V., “Backlund Transformations Between Four Lax-Integrable 3D Equations”, J. Nonlinear Math. Phys., 24:4 (2017), 465–468  crossref  mathscinet  isi  scopus
    11. Akhmedova V. Takebe T. Zabrodin A., “Multi-Variable Reductions of the Dispersionless DKP Hierarchy”, J. Phys. A-Math. Theor., 50:48 (2017), 485204  crossref  mathscinet  zmath  isi  scopus
    12. Doubrov B. Ferapontov E.V. Kruglikov B. Novikov V.S., “On Integrability in Grassmann Geometries: Integrable Systems Associated With Fourfolds in Gr(3,5)”, Proc. London Math. Soc., 116:5 (2018), 1269–1300  crossref  mathscinet  zmath  isi  scopus
    13. M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105  mathnet  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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