Advertisement

Theoretical and Mathematical Physics

, Volume 154, Issue 2, pp 209–219 | Cite as

Classification of integrable Vlasov-type equations

  • A. V. OdesskiiEmail author
  • M. V. Pavlov
  • V. V. Sokolov
Article

Abstract

The classification of integrable Vlasov-type equations reduces to a functional equation for a generating function. We find a general solution of this functional equation in terms of hypergeometric functions.

Keywords

integrable hydrodynamic chain hydrodynamic reduction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. E. Zakharov, Funct. Anal. Appl., 14, 89–98 (1980); V. E. Zakharov, Phys. D, 3, 193–202 (1981).zbMATHCrossRefGoogle Scholar
  2. 2.
    M. V. Pavlov, J. Phys. A, 39, 10803–10819 (2006).zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    E. V. Ferapontov and D. G. Marshall, Math. Ann., 339, 61–99 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. V. Pavlov, Comm. Math. Phys., 272, 469–505 (2007).zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    D. J. Benney, Stud. Appl. Math., 52, 45–50 (1973).zbMATHGoogle Scholar
  6. 6.
    J. Gibbons, Phys. D, 3, 503–511 (1981).CrossRefMathSciNetGoogle Scholar
  7. 7.
    B. A. Kupershmidt and Yu. I. Manin, Funct. Anal. Appl., 11, 188–197 (1977); 12, 20–29 (1978).CrossRefMathSciNetGoogle Scholar
  8. 8.
    B. A. Kupershmidt, Proc. Roy. Irish Acad. Sec. A, 83, 45–74 (1983); “Normal and universal forms in integrable hydrodynamical systems,” in: Proc. Berkeley-Ames Conference on Nonlinear Problems in Control and Fluid Dynamics, Vol. 2, Lie Groups: Hist. Frontiers Appl. Ser. B: Systems Inform. Control, Math Sci. Press, Brookline, Mass. (1984), pp. 357–378.zbMATHMathSciNetGoogle Scholar
  9. 9.
    M. V. Pavlov, Internat. Math. Res. Notices, Article ID 46987 (2006).Google Scholar
  10. 10.
    E. V. Ferapontov and K. R. Khusnutdinova, Comm. Math. Phys., 248, 187–206 (2004); J. Phys. A, 37, 2949–2963 (2004).zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    J. Gibbons and S. P. Tsarev, Phys. Lett. A, 211, 19–24 (1996); 258, 263–271 (1999).zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    S. P. Tsarev, Sov. Math. Dokl., 31, 488–491 (1985); Math. USSR Izv., 37, 397–419 (1991).zbMATHGoogle Scholar
  13. 13.
    M. V. Pavlov, Theor. Math. Phys., 138, 45–58 (2004).CrossRefGoogle Scholar
  14. 14.
    A. V. Odesskii and V. V. Sokolov, “On (2+1)-dimensional systems of hydrodynamic type possessing a pseudopotential with movable singularities,” Funct. Anal. Appl., 42 (to appear, 2008).Google Scholar
  15. 15.
    A. V. Odesskii, “A family of (2+1)-dimensional hydrodynamic type systems possessing pseudopotential,” Selecta Math. (to appear); arXiv:0704.3577v3 [math.AP] (2007).Google Scholar
  16. 16.
    I. M. Krichever, Comm. Pure Appl. Math., 47, 437–475 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    B. A. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620), Springer, Berlin (1996), pp. 120–348.CrossRefGoogle Scholar
  18. 18.
    A. V. Zabrodin, Theor. Math. Phys., 129, 1511–1525.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • A. V. Odesskii
    • 1
    • 2
    Email author
  • M. V. Pavlov
    • 3
  • V. V. Sokolov
    • 1
  1. 1.Landau Institute for Theoretical PhysicsRASMoscowRussia
  2. 2.University of ManchesterManchesterUK
  3. 3.Lebedev Physical InstituteRASMoscowRussia

Personalised recommendations