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

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:

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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• http://mi.mathnet.ru/eng/tmf/v163/i2/p179

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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.
2. Odesskii A.V. Sokolov V.V., “Non-Homogeneous Systems of Hydrodynamic Type Possessing Lax Representations”, Commun. Math. Phys., 324:1 (2013), 47–62
3. Habibullin I., “Characteristic Lie Rings, Finitely-Generated Modules and Integrability Conditions for (2+1)-Dimensional Lattices”, Phys. Scr., 87:6 (2013), 065005
4. Takashi Takebe, “Dispersionless BKP Hierarchy and Quadrant Löwner Equation”, SIGMA, 10 (2014), 023, 13 pp.
5. Ferapontov E.V. Kruglikov B.S., “Dispersionless Integrable Systems in 3D and Einstein-Weyl Geometry”, J. Differ. Geom., 97:2 (2014), 215–254
6. Ferapontov E.V. Moss J., “Characteristic Integrals in 3D and Linear Degeneracy”, J. Nonlinear Math. Phys., 21:2 (2014), 214–224
7. Ferapontov E.V. Moss J., “Linearly Degenerate Partial Differential Equations and Quadratic Line Complexes”, Commun. Anal. Geom., 23:1 (2015), 91–127
8. A. V. Odesskii, “Integrable structures of dispersionless systems and differential geometry”, Theoret. and Math. Phys., 191:2 (2017), 692–709
9. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.
10. Morozov O.I. Pavlov M.V., “Backlund Transformations Between Four Lax-Integrable 3D Equations”, J. Nonlinear Math. Phys., 24:4 (2017), 465–468
11. Akhmedova V. Takebe T. Zabrodin A., “Multi-Variable Reductions of the Dispersionless DKP Hierarchy”, J. Phys. A-Math. Theor., 50:48 (2017), 485204
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
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
14. Ufa Math. J., 11:3 (2019), 109–131
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