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TMF, 2007, Volume 151, Number 3, Pages 439–457 (Mi tmf6058)  

This article is cited in 17 scientific papers (total in 17 papers)

Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

B. G. Konopelchenkoa, F. Magrib

a Lecce University
b Universita di Milano-Bicocca

Abstract: We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level.

Keywords: associative algebra, coisotropic deformation, integrable equation

DOI: https://doi.org/10.4213/tmf6058

Full text: PDF file (418 kB)
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English version:
Theoretical and Mathematical Physics, 2007, 151:3, 803–819

Bibliographic databases:


Citation: B. G. Konopelchenko, F. Magri, “Dispersionless integrable equations as coisotropic deformations: Extensions and reductions”, TMF, 151:3 (2007), 439–457; Theoret. and Math. Phys., 151:3 (2007), 803–819

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. Szablikowski B.M., Blaszak M., “Dispersionful analog of the Whitham hierarchy”, J. Math. Phys., 49:8 (2008), 082701, 20 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. B. G. Konopelchenko, “Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras”, Theoret. and Math. Phys., 159:3 (2009), 842–852  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Konopelchenko B.G., “Quantum deformations of associative algebras and integrable systems”, J. Phys. A, 42:9 (2009), 095201, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Manakov S.V., Santini P.M., “The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behaviour, implicit solutions and wave breaking”, J. Phys. A, 42:9 (2009), 095203, 16 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. V. A. Belavin, “Modular integrals in minimal super Liouville gravity”, Theoret. and Math. Phys., 161:1 (2009), 1361–1375  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. Konopelchenko B.G., “Discrete integrable systems and deformations of associative algebras”, J. Phys. A, 42:45 (2009), 454003, 35 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Sergyeyev A., “Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations”, J. Phys. A, 42:40 (2009), 404017, 15 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    8. Konopelchenko B.G., Ortenzi G., “Coisotropic deformations of algebraic varieties and integrable systems”, J. Phys. A, 42:41 (2009), 415207, 18 pp.  crossref  mathscinet  zmath  isi  scopus
    9. Manakov S.V., Santini P.M., “On the solutions of the second heavenly and Pavlov equations”, J. Phys. A, 42:40 (2009), 404013, 11 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    10. Konopelchenko B.G., “On the Deformation Theory of Structure Constants for Associative Algebras”, Adv. Math. Phys., 2010, 389091  crossref  mathscinet  zmath  isi  scopus
    11. Manakov S.V., Santini P.M., “Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking”, J. Phys. A: Math. Theor., 44:34 (2011), 345203  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. B. G. Konopelchenko, G. Ortenzi, “Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian”, Theoret. and Math. Phys., 177:2 (2013), 1479–1491  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Konopelchenko B.G., Ortenzi G., “Birkhoff Strata of Sato Grassmannian and Algebraic Curves”, J. Nonlinear Math. Phys., 20:3 (2013), 309–347  crossref  mathscinet  isi  scopus
    14. Pavlov M.V. Sergyeyev A., “Oriented Associativity Equations and Symmetry Consistent Conjugate Curvilinear Coordinate Nets”, J. Geom. Phys., 85 (2014), 46–59  crossref  mathscinet  zmath  adsnasa  isi  scopus
    15. Manakov S.V. Santini P.M., “Integrable Dispersionless PDEs Arising as Commutation Condition of Pairs of Vector Fields”, Physics and Mathematics of Nonlinear Phenomena 2013, Journal of Physics Conference Series, 482, IOP Publishing Ltd, 2014, 012029  crossref  isi  scopus
    16. Grinevich P.G., Santini P.M., Wu D., “the Cauchy Problem For the Pavlov Equation”, Nonlinearity, 28:11 (2015), 3709–3754  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Prykarpatski A.K., “On the Solutions to the Witten-Dijkgraaf-Verlinde-Verlinde Associativity Equations and Their Algebraic Properties”, J. Geom. Phys., 134 (2018), 77–83  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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