Skip to main content
SpringerLink
Log in

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

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

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.”

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alan C. Newell, Solitons in Mathematics and Physics (CBMS-NSF Reg. Conf. Ser. Appl. Math., Vol. 48), SIAM, Philadelphia, Pa. (1985).

    Google Scholar 

  2. L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach to the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtadzhyan, The Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).

    Google Scholar 

  3. V. E. Zakharov, ed., What is Integrability? Springer, Berlin (1991).

    Google Scholar 

  4. A. V. Mikhailov, Integrability (Lect. Notes Phys., Vol. 767), Springer, Berlin (2009).

    Book  MATH  Google Scholar 

  5. E. V. Zakharov, “Dispersionless limit of integrable systems in 2+1 dimensions,” in: Singular Limits of Dispersive Waves (NATO ASI 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 

  6. I. M. Krichever, Comm. Pure Appl. Math., 47, 437–475 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. V. Odesskii and V. V. Sokolov, Funct. Anal. Appl., 42, No. 3, 205–212 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 13, No. 3, 166–174 (1979).

    MATH  MathSciNet  Google Scholar 

  9. M. V. Saveliev and A. M. Vershik, Comm. Math. Phys., 126, 367–378 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. S. V. Manakov and P. M. Santini, Phys. Lett. A, 359, 613–619 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. S. V. Manakov and P. M. Santini, Theor. Math. Phys., 152, 1004–1011 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Gibbons and S. P. Tsarev, Phys. Lett. A, 211, 19–24 (1996).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. E. V. Ferapontov and K. R. Khusnutdinova, Comm. Math. Phys., 248, 187–206 (2004).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. E. V. Ferapontov and K. R. Khusnutdinova, J. Phys. A, 37, 2949–2963 (2004).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. E. V. Ferapontov and K. R. Khusnutdinova, J. Math. Phys., 45, 2365–2377 (2004).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. A. V. Odesskii, Selecta Math., 13, 727–742 (2008); arXiv:0704.3577v3 [math.AP] (2007).

    Article  MathSciNet  Google Scholar 

  17. E. V. Ferapontov, K. R. Khusnutdinova, and S. P. Tsarev, Comm. Math. Phys., 261, 225–243 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. E. V. Ferapontov and A. V. Odesskii, “Integrable Lagrangians and modular forms,” arXiv:0707.3433v2 [nlin.SI] (2007).

  19. P. A. Burovskiy, E. V. Ferapontov, and S. P. Tsarev, “Second order quasilinear PDEs and conformal structures in projective space,” arXiv:0802.2626v3 [nlin.SI] (2008).

  20. A. V. Odesskii and V. V. Sokolov, “Integrable pseudopotentials related to generalized hypergeometric functions,” arXiv:0803.0086v3 [nlin.SI] (2008).

  21. I. M. Gel’fand, M. I. Graev, and V. S. Retakh, Russ. Math. Surveys, 47, 1–88 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  22. E. V. Ferapontov, L. Hadjikos, and K. R. Khusnutdinova, “Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian,” arXiv:0705.1774v1 [math.DG] (2007).

  23. M. V. Pavlov, Theor. Math. Phys., 138, 45–58 (2004).

    Article  MATH  Google Scholar 

  24. M. V. Pavlov, J. Phys. A, 39, 10803–10819 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. E. V. Ferapontov and D. G. Marshal, Math. Ann., 339, 61–99 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  26. I. M. Krichever, Comm. Math. Phys., 143, 415–429 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. A. V. Odesskii, M. V. Pavlov, and V. V. Sokolov, Theor. Math. Phys., 154, 209–219 (2008); arXiv:0710.5655v2 [nlin.SI] (2007).

    Article  MathSciNet  Google Scholar 

  28. A. V. Odesskii and V. V. Sokolov, Theor. Math. Phys., 161, 1340–1352 (2009); arXiv:0810.3879v1 [nlin.SI] (2008).

    Article  MATH  Google Scholar 

  29. V. P. Spiridonov, Russ. Math. Surveys, 63, 405–472 (2008).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. K. Löwner, Math. Ann., 89, 103–121 (1923).

    Article  MATH  MathSciNet  Google Scholar 

  31. K. Takasaki and T. Takebe, Rev. Math. Phys., 7, 743–808 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  32. V. Shramchenko, J. Phys. A, 36, 10585–10605 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. T. Takebe, L.-P. Leo, and A. Zabrodin, J. Phys. A, 39, 11479–11501 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. M. Mañas, E. Medina, and L. Martínes Alonso, J. Phys. A, 39, 2349–2381 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  35. A. V. Mikhailov and V. V. Sokolov, “Symmetries of differential equations and the problem of integrability,” in: Integrability (Lect. Notes Phys., Vol. 767, A. V. Mikhailov, ed.), Springer, Berlin (2009), pp. 19–88.

    Chapter  Google Scholar 

  36. A. V. Mikhailov and R. I. Yamilov, J. Phys. A, 31, 6707–6715 (1998).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys, 44, 35–124 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. S. P. Tsarëv, Math. USSR-Izv., 37, 397–419 (1991).

    Article  MathSciNet  Google Scholar 

  39. S. P. Tsarëv, Sov. Math. Dokl., 31, 488–491 (1985).

    MATH  Google Scholar 

  40. E. V. Ferapontov, Phys. Lett. A, 158, 112–118 (1991).

    Article  MathSciNet  ADS  Google Scholar 

  41. M. V. Pavlov, J. Math. Phys., 44, 4134–4156 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. M. V. Pavlov, Comm. Math. Phys., 272, 469–505 (2007).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. E. V. Ferapontov, K. R. Khusnutinova, and C. Klein, “On linear degeneracy of integrable quasilinear systems in higher dimensions,” arXiv:0909.5685v2 [nlin.SI] (2009).

  44. A. V. Odesskii and V. V. Sokolov, “Systems of Gibbons-Tsarev type and integrable 3-dimensional models,” arXiv:0906.3509v1 [nlin.SI] (2009).

  45. D. J. Benney, Stud. Appl. Math., 52, 45–50 (1973).

    MATH  Google Scholar 

  46. B. A. Kupershmidt and Yu. I. Manin, Funct. Anal. Appl., 11, No. 3, 188–197 (1977).

    Article  MathSciNet  Google Scholar 

  47. V. E. Zakharov, Phys. D, 3, 193–202 (1981).

    Article  Google Scholar 

  48. L. Martínez Alonso and A. B. Shabat, Theor. Math. Phys., 140, 1073–1085 (2004).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Landau Institute for Theoretical Physics, RAS, Moscow, Russia

    A. V. Odesskii & V. V. Sokolov

  2. Brock University, St. Catharines, Ontario, Canada

    A. V. Odesskii

Authors
  1. A. V. Odesskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Sokolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Odesskii.

Additional information

This paper was written at the request of the Editorial Board.

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 163, No. 2, pp. 179–221, May, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Odesskii, A.V., Sokolov, V.V. Integrable (2+1)-dimensional systems of hydrodynamic type. Theor Math Phys 163, 549–586 (2010). https://doi.org/10.1007/s11232-010-0043-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-010-0043-1

Keywords

Search

Navigation