Skip to main content
SpringerLink
Log in

LÖwner evolution and finite-dimensional reductions of integrable systems

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

Abstract

The Löwner equation is known as the one-dimensional reduction of the Benney chain and also as the dispersionless KP hierarchy. We propose a reverse process and show that time splitting in the Löwner or the Löwner-Kufarev equation leads to some known integrable systems.

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. Y. Kodama and J. Gibbons, Phys. Lett. A, 135, 167–170 (1989).

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. E. A. Zabolotskaya and R. V. Khokhlov, Sov. Phys. Acoust., 15, 35–40 (1969).

    Google Scholar 

  4. V. E. Zakharov, Funct. Anal. Appl., 14, 89–98 (1980).

    Article  MATH  Google Scholar 

  5. J. Gibbons and S. Tsarev, Phys. Lett. A, 258, 263–271 (1999).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. M. Mañas, L. Martínez Alonso, and E. Medina, J. Phys. A, 35, 401–417 (2002).

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. B. Gustafsson and A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser, Basel (2006).

    MATH  Google Scholar 

  9. S. Richardson, J. Fluid. Mech., 56, 609–618 (1972).

    Article  ADS  MATH  Google Scholar 

  10. M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, Phys. Rev. Lett., 84, 5106–5109 (2000).

    Article  ADS  Google Scholar 

  11. I. Markina and A. Vasil’ev, “Virasoro algebra and dynamics in the space of univalent functions,” in: Five Lectures in Complex Analysis (Contemp. Math., Vol. 525, M. D. Contreras and S. Diaz-Madrigal, eds.), Amer. Math. Soc., Providence, R. I. (2010), pp. 85–116.

    Chapter  Google Scholar 

  12. M. V. Pavlov, Internat. Math. Res. Notices, 2006, 46987 (2006).

    Google Scholar 

  13. A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, New York (1961).

    Google Scholar 

  14. M. V. Pavlov and S. P. Tsarev, “Classical mechanical systems with one-and-a-half degrees of freedom and Vlasov kinetic equation,” Amer. Math. Soc. Transl. Ser. 2 (to appear); arXiv:1306.3737v2 [nlin.SI] (2013).

    Google Scholar 

  15. B. A. Kupershmidt and Yu. I. Manin, Funct. Anal. Appl., 11, 188–197 (1977).

    Article  MathSciNet  Google Scholar 

  16. M. V. Pavlov, Commun. Math. Phys., 272, 469–505 (2007).

    Article  ADS  MATH  Google Scholar 

  17. J. Gibbons, Phys. Lett. A, 90, 7–8 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  18. O. Tammi, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 13, 125–136 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  19. M. V. Pavlov and S. P. Tsarev, Russ. Math. Surveys, 4, 196–197 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  20. B. A. Kupershmidt and Yu. I. Manin, Funct. Anal. Appl., 12, 20–29 (1978).

    MATH  MathSciNet  Google Scholar 

  21. S. P. Tsarev, Soviet Math. Dokl., 31, 488–491 (1985).

    MATH  Google Scholar 

  22. S. P. Tsarev, Math. USSR-Izv., 37, 397–419 (1991).

    Article  MathSciNet  Google Scholar 

  23. E. V. Ferapontov, Private communication.

  24. B. A. Kupershmidt, Proc. Roy. Irish Acad. Sect. A, 83, 45–74 (1983); “Normal and universal forms in integrable hydrodynamical systems,” in: Proceedings of Berkeley-Ames Conference on Nonlinear Problems in Control and Fluid Dynamics (Berkeley, Calif., 1983, L. R. Hunt and C. F. Martin, eds.), Math. Sci. Press, Brookline, Mass. (1984), pp. 357–378.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lebedev Physical Institute, RAS, Moscow, Russia

    M. V. Pavlov

  2. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia

    M. V. Pavlov

  3. Saratov State University, Saratov, Russia

    D. V. Prokhorov & A. M. Zakharov

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    A. Yu. Vasil’ev

Authors
  1. M. V. Pavlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. V. Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Yu. Vasil’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. M. Zakharov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Pavlov.

Additional information

Dedicated to Ludvig D. Faddeev on the occasion of his 80th birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pavlov, M.V., Prokhorov, D.V., Vasil’ev, A.Y. et al. LÖwner evolution and finite-dimensional reductions of integrable systems. Theor Math Phys 181, 1263–1278 (2014). https://doi.org/10.1007/s11232-014-0211-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-014-0211-9

Keywords

Search

Navigation