Skip to main content
SpringerLink
Log in

Two-dimensional rational solitons and their blowup via the moutard transformation

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

Abstract

We construct a family of two-dimensional stationary Schrödinger operators with rapidly decaying smooth rational potentials and nontrivial L2 kernels. We show that some of the constructed potentials generate solutions of the Veselov-Novikov equation that decay rapidly at infinity, are nonsingular at t = 0, and have singularities at finite times t ≥ t0 > 0.

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.

Similar content being viewed by others

References

  1. T. Moutard, J. École Polytec., 45, 1–11 (1878).

    Google Scholar 

  2. I. A. Taimanov and S. P. Tsarev, Russ. Math. Surveys, 62, 631–633 (2007).

    Article  MathSciNet  Google Scholar 

  3. L. D. Faddeev, J. Sov. Math., 5, 334–396 (1974).

    Article  Google Scholar 

  4. R. G. Novikov and G. M. Khenkin, Russ. Math. Surveys, 42, No. 3, 109–180 (1987).

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  6. C. E. Kenig, G. Ponce, and L. Vega, J. Amer. Math. Soc., 4, 323–347 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  7. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Sov. Math. Dokl., 17, 947–952 (1976).

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Darboux, C. R. Acad. Sci. Paris, 94, 1456–1459 (1882).

    Google Scholar 

  11. M. Adler and J. Moser, Comm. Math. Phys., 61, 1–30 (1978).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. H. Airault, H. P. McKean, and J. Moser, Comm. Pure Appl. Math., 30, 95–148 (1977).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. S. P. Novikov and A. P. Veselov, “Exactly solvable two-dimensional Schrödinger operators and Laplace transformations,” in: Solitons, Geometry, and Topology: On the Crossroad (Amer. Math. Soc. Transl. Ser. 2, Vol. 179, V. M. Buchstaber and S. P. Novikov, eds.), Amer. Math. Soc., Providence, R. I. (1997), pp. 109–132.

    Google Scholar 

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

    MATH  Google Scholar 

  15. H.-C. Hu, S. Y. Lou, and Q.-P. Liu, Chinese Phys. Lett., 20, 1413–1415 (2003).

    Article  ADS  Google Scholar 

  16. C. Athorne and J. J. C. Nimmo, Inverse Probl., 7, 809–826 (1991).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. B. A. Dubrovin and S. P. Novikov, JETP, 40, 1058–1063 (1974).

    ADS  MathSciNet  Google Scholar 

  18. L. Bianchi, Lezioni di geometria differenziale, Vols. 1–4, Zanichelli, Bologna (1923–1927).

    Google Scholar 

  19. E. I. Ganzha and S. P. Tsarev, Russ. Math. Surveys, 51, 1200–1202 (1996); arXiv:solv-int/9606003v2 (1996).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Novosibirsk, Russia

    I. A. Taimanov

  2. Krasnoyarsk State Pedagogical University, Krasnoyarsk, Russia

    S. P. Tsarev

Authors
  1. I. A. Taimanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. P. Tsarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. A. Taimanov.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 188–207, November, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taimanov, I.A., Tsarev, S.P. Two-dimensional rational solitons and their blowup via the moutard transformation. Theor Math Phys 157, 1525–1541 (2008). https://doi.org/10.1007/s11232-008-0127-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-008-0127-3

Keywords

Search

Navigation