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TMF, 2004, Volume 138, Number 1, Pages 55–70 (Mi tmf12)  

This article is cited in 26 scientific papers (total in 26 papers)

Classifying Integrable Egoroff Hydrodynamic Chains

M. V. Pavlov

Loughborough University

Abstract: We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.

Keywords: hydrodynamic chains and lattices, Egoroff integrable systems, dispersionless Hirota equations, tau function


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Theoretical and Mathematical Physics, 2004, 138:1, 45–58

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Received: 15.05.2003

Citation: M. V. Pavlov, “Classifying Integrable Egoroff Hydrodynamic Chains”, TMF, 138:1 (2004), 55–70; Theoret. and Math. Phys., 138:1 (2004), 45–58

Citation in format AMSBIB
\by M.~V.~Pavlov
\paper Classifying Integrable Egoroff Hydrodynamic Chains
\jour TMF
\yr 2004
\vol 138
\issue 1
\pages 55--70
\jour Theoret. and Math. Phys.
\yr 2004
\vol 138
\issue 1
\pages 45--58

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    This publication is cited in the following articles:
    1. Ferapontov E.V., Khusnutdinova K.R., “The characterization of two-component $(2+1)$-dimensional integrable systems of hydrodynamic type”, J. Phys. A, 37:8 (2004), 2949–2963  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. E. V. Ferapontov, K. R. Khusnutdinova, M. V. Pavlov, “Classification of Integrable $(2+1)$-Dimensional Quasilinear Hierarchies”, Theoret. and Math. Phys., 144:1 (2005), 907–915  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Chang Jen-Hsu, “Hydrodynamic reductions of the dispersionless Harry Dym hierarchy”, J. Phys. A, 38:29 (2005), 6505–6515  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. Chen Yu-Tung, Tu Ming-Hsien, “On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy”, J. Math. Phys., 47:10 (2006), 102702, 19 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. Chang Jen-Hsu, “On the waterbag model of dispersionless KP hierarchy”, J. Phys. A, 39:36 (2006), 11217–11230  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    6. Pavlov M.V., “Classification of integrable hydrodynamic chains and generating functions of conservation laws”, J. Phys. A, 39:34 (2006), 10803–10819  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. Ferapontov E.V., Khusnutdinova K.R., “The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462:2068 (2006), 1197–1219  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Ferapontov E.V., Khusnutdinova K.R., Tsarev S.P., “On a class of three-dimensional integrable Lagrangians”, Comm. Math. Phys., 261:1 (2006), 225–243  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    9. Pavlov M.V., “Hydrodynamic chains and the classification of their Poisson brackets”, J. Math. Phys., 47:12 (2006), 123514, 15 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    10. Pavlov M.V., “The Kupershmidt hydrodynamic chains and lattices”, Int. Math. Res. Not., 2006, 46987, 43 pp.  mathscinet  zmath  isi  elib
    11. Ferapontov E.V., Marshall D.G., “Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor”, Math. Ann., 339:1 (2007), 61–99  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. A. V. Odesskii, M. V. Pavlov, V. V. Sokolov, “Classification of integrable Vlasov-type equations”, Theoret. and Math. Phys., 154:2 (2008), 209–219  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. Ferapontov E.V., Moro A., Sokolov V.V., “Hamiltonian systems of hydrodynamic type in $2+1$ dimensions”, Comm. Math. Phys., 285:1 (2009), 31–65  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    14. Ferapontov E.V., Hadjikos L., Khusnutdinova K.R., “Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian”, Int. Math. Res. Not. IMRN, 2010, no. 3, 496–535  crossref  mathscinet  zmath  isi  scopus
    15. 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
    16. A. V. Odesskii, V. V. Sokolov, “Integrable $(2+1)$-dimensional systems of hydrodynamic type”, Theoret. and Math. Phys., 163:2 (2010), 549–586  mathnet  crossref  crossref  adsnasa  isi  elib
    17. 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
    18. Burovskiy P.A., Ferapontov E.V., Tsarev S.P., “Second-order quasilinear PDEs and conformal structures in projective space”, Internat. J. Math., 21:6 (2010), 799–841  crossref  mathscinet  zmath  isi  elib  scopus
    19. Oleg I. Morozov, “A two-component generalization of the integrable rdDym equation”, SIGMA, 8 (2012), 051, 5 pp.  mathnet  crossref  mathscinet
    20. 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
    21. Y. Kodama, B. G. Konopelchenko, “Confluence of hypergeometric functions and integrable hydrodynamic-type systems”, Theoret. and Math. Phys., 188:3 (2016), 1334–1357  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    22. Prykarpatski A.K., Hentosh O.E., Prykarpatsky Ya.A., “Geometric Structure of the Classical Lagrange–dAlembert Principle and Its Application to Integrable Nonlinear Dynamical Systems”, 5, no. 4, 2017, 75  crossref  zmath  isi  scopus
    23. Hentosh O.E. Prykarpatsky Ya.A. Blackrnore D. Prykarpatski A.K., “Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange–dAlembert principle”, J. Geom. Phys., 120 (2017), 208–227  crossref  mathscinet  zmath  isi  scopus
    24. Z. V. Makridin, “An effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type”, Theoret. and Math. Phys., 194:2 (2018), 274–283  mathnet  crossref  crossref  adsnasa  isi  elib
    25. 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
    26. Prykarpatskyy Ya.A. Samoilenko A.M., “Classical M. a. Buhl Problem, Its Pfeiffer–Sato Solutions, and the Classical Lagrange–dAlembert Principle For the Integrable Heavenly-Type Nonlinear Equations”, Ukr. Math. J., 69:12 (2018), 1924–1967  crossref  mathscinet  isi  scopus
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