Skip to main content
SpringerLink
Log in

New exact solutions with functional parameters of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity

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

Abstract

We use the Zakharov—Manakov δ-dressing method to construct new classes of exact solutions with functional parameters of the hyperbolic and elliptic versions of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity. We show that the constructed solutions contain classes of multisoliton solutions, which at a fixed time are exact potentials of the perturbed telegraph equation (the perturbed string equation) and the two-dimensional stationary Schrödinger equation. We interpret the stationary states of a microparticle in soliton-type potential fields physically in accordance with the constructed exact wave functions for the two-dimensional stationary Schrödinger equation.

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. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transi.: S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Plenum, New York (1984).

    MATH  Google Scholar 

  2. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (London Math. Soc. Lect. Note Ser., Vol. 149), Cambridge Univ. Press, Cambridge (1991).

    Book  MATH  Google Scholar 

  3. B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations: The Inverse Spectral Transform in 2+1 Dimensions, Plenum, New York (1992).

    MATH  Google Scholar 

  4. B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectral Transform Method, World Scientific, Singapore (1993).

    MATH  Google Scholar 

  5. S. V. Manakov, Phys. D, 3, 420–427 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Beals and R. R. Coifman, Phys. D, 18, 242–249 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. E. Zakharov and S. V. Manakov, Funct. Anal Appl, 19, No. 2, 89–101 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. E. Zakharov, “Commutating operators and nonlocal problem,” in: Nonlinear and Turbulent Processes in Physics, Vol. 1, Naukova Dumka, Kiev (1988), pp. 152–154.

    Google Scholar 

  9. L. V. Bogdanov and S. V. Manakov, J. Phys. A, 21, L537–L544 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  10. A. S. Fokas and V. E. Zakharov, J. Nonlinear Sci., 2, 109–134 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. L. P. Nizhnik, Sov. Phys. Dokl, 25, 706–708 (1980).

    MATH  ADS  Google Scholar 

  12. A. P. Veselov and S. P. Novikov, Sov. Math. Dokl, 30, 588–591 (1984).

    MATH  Google Scholar 

  13. P. G. Grinevich and S. V. Manakov, Funct. Anal Appl, 20, No. 2, 103–113 (1986).

    Article  MathSciNet  Google Scholar 

  14. P. G. Grinevich and S. P. Novikov, “Inverse scattering problem for the two-dimensional Schrodinger operator at a fixed negative energy and generalized analytic functions,” in: Plasma Theory and Nonlinear and Turbulent Processes in Physics (Proc. Intl. Workshop “Plasma Theory and Nonlinear and Turbulent Processes in Physics,” V. G. Bar’yakhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, eds.). Vol. 1, World Scientific, Singapore (1988), pp. 58–85.

    Google Scholar 

  15. P. G. Grinevich and S. P. Novikov, Funct. Anal Appl, 22, No. 1, 19–27 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Boiti, J. Jp. Leon, M. Manna, and F. Pempinelli, Inverse Problems, 3, 25–36 (1987).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).

    MATH  Google Scholar 

  18. P. G. Grinevich, “Scattering transform for the two-dimensional Schrödinger equation with one energy and integrable equations of mathematical physics related to it [in Russian],” Doctoral dissertation, Landau Inst. Theor. Phys., RAS, Chernogolovka (1999).

    Google Scholar 

  19. P. G. Grinevich, Russ. Math. Surveys, 55, 1015–1083 (2000).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. C. Athorne and J. J. C. Nimmo, Inverse Problems, 7, 809–826 (1991).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. V. G. Dubrovsky and B. G. Konopelchenko, Inverse Problems, 9, 391–416 (1993).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. V. G. Dubrovsky and I. B. Formusatik, J. Phys. A, 34, 1837–1851 (2001).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. V. G. Dubrovsky and I. B. Formusatik, Phys. Lett. A, 313, 68–76 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. S. V. Manakov, Russ. Math. Surveys, 31, 245–246 (1976).

    MATH  MathSciNet  Google Scholar 

  25. P. G. Grinevich, Theor. Math. Phys., 69, 1170–1172 (1986).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Sov. Math. Dokl, 17, 947–951 (1976, 1977).

    MATH  Google Scholar 

  27. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 8, No. 3, 226–235 (1974).

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. V. G. Dubrovsky and A. V. Gramolin, J. Phys. A, 41, 275208 (2008); arXiv:0802.2334v2 [nlin.SI] (2008).

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State University for Technology, Novosibirsk, Russia

    V. G. Dubrovsky, A. V. Topovsky & M. Yu. Basalaev

Authors
  1. V. G. Dubrovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Topovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Yu. Basalaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Dubrovsky.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 165, No. 2, pp. 272–294, November, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dubrovsky, V.G., Topovsky, A.V. & Basalaev, M.Y. New exact solutions with functional parameters of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity. Theor Math Phys 165, 1470–1489 (2010). https://doi.org/10.1007/s11232-010-0122-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-010-0122-3

Keywords

Search

Navigation