Integrable elliptic pseudopotentials



We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.


integrable three-dimensional system of hydrodynamic type elliptic hypergeometric function 


  1. 1.
    A. Odesskii and V. Sokolov, “Integrable pseudopotentials related to generalized hypergeometric functions,” arXiv:0803.0086v3 [nlin.SI] (2008).Google Scholar
  2. 2.
    I. M. Gel’fand, M. I. Graev, and V. S. Retakh, Russ. Math. Surveys, 47, No. 4, 1–88 (1992).CrossRefMathSciNetGoogle Scholar
  3. 3.
    V. E. Zakharov, “Dispersionless limit of integrable systems in 2+1 dimensions,” in: Singular Limits of Dispersive Waves (NATO Adv. Sci. Inst. Ser. B Phys., Vol. 320, N. M. Ercolani, I. R. Gabitov, C. D. Levermore, and D. Serre, eds.), Plenum, New York (1994), pp. 165–174.Google Scholar
  4. 4.
    I. M. Krichever, Comm. Math. Phys., 143, 415–429 (1992).MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 13, 166–174 (1979).MATHMathSciNetGoogle Scholar
  6. 6.
    I. M. Krichever, Comm. Pure Appl. Math., 47, 437–475 (1994).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    B. A. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620, M. Francaviglia and S. Greco, eds.), Springer, Berlin (1996), pp. 120–348.CrossRefGoogle Scholar
  8. 8.
    A. V. Odesskii, Selecta Math., 13, 727–742 (2008).CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. P. Spiridonov, Russ. Math. Surveys, 63, 405–472 (2008).MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    J. Gibbons and S. P. Tsarev, Phys. Lett. A, 211, 19–24 (1996); 258, 263–271 (1999).MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    E. V. Ferapontov and K. R. Khusnutdinova, Comm. Math. Phys., 248, 187–206 (2004); J. Phys. A, 37, 2949–2963 (2004).MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys, 44, 35–124 (1989).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    S. P. Tsarev, Sov. Math. Dokl., 31, 488–491 (1985); Math. USSR-Izv., 37, 397–419 (1991).MATHGoogle Scholar
  14. 14.
    M. V. Pavlov, Comm. Math. Phys., 272, 469–505 (2007).MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    V. A. Shramchenko, J. Phys. A, 36, 10585–10605 (2003); arXiv:math-ph/0402014v1 (2004).MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    A. A. Akhmetshin, Yu. S. Vol’vovskii, and I. M. Krichever, Russ. Math. Surveys, 54, 427–429 (1999).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    E. V. Ferapontov and D. G. Marshal, Math. Ann., 339, 61–99 (2007).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    M. V. Pavlov, J. Phys. A, 39, 10803–10819 (2006).MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    A. V. Odesskii, M. V. Pavlov, and V. V. Sokolov, Theor. Math. Phys., 154, 209–219 (2008).CrossRefMathSciNetGoogle Scholar

Copyright information

© MAIK/Nauka 2009

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia
  2. 2.Brock UniversitySt. CatharinesCanada

Personalised recommendations