Skip to main content
SpringerLink
Log in

The AKNS Hierarchy and Finite-Gap Schrödinger Potentials

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

Abstract

We consider the AKNS hierarchy and find the necessary and sufficient conditions for functions p and q to become solutions of the AKNS hierarchy. Using the functions p and q, we construct finite-gap Schrödinger potentials.

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. G. Wallenberg, “Über die Vertauschbarkeit homogener linearer Differentialausdrücke,” Arch. Math. Phys. (3), 4, 252–268 (1903).

    MATH  Google Scholar 

  2. J. Schur, “Über vertauschbare lineare Differentialausdrücke,” Sitzungsber. Berl. Math. Ges., 4, 2–8 (1905).

    MATH  Google Scholar 

  3. J. L. Burchnall and T. W. Chaundy, “Commutative ordinary differential operators,” Proc. London Math. Soc., 21, 420–440 (1923).

    Article  MathSciNet  MATH  Google Scholar 

  4. P. G. Grinevich, “Vector rank of commuting matrix differential operators: Proof of S. P. Novikov’s criterion,” Math. USSR-Izv., 28, 445–465 (1987).

    Article  MATH  Google Scholar 

  5. J. Dixmier, “Sur les algèbres de Weyl,” Bull. Soc. Math. France, 96, 209–242 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  6. I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funct. Anal. Appl., 11, 12–26 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  7. I. M. Krichever, “Commutative rings of ordinary linear differential operators,” Funct. Anal. Appl., 12, 175–185 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  8. I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations,” Russ. Math. Surveys, 35, 53–79 (1980).

    Article  ADS  MATH  Google Scholar 

  9. O. I. Mokhov, “Commuting ordinary differential operators of rank 3 corresponding to an elliptic curve,” Russ. Math. Surveys, 37, 129–130 (1982).

    Article  ADS  MATH  Google Scholar 

  10. O. I. Mokhov, “Commuting differential operators of rank 3, and nonlinear differential equations,” Math. USSRIzv., 35, 629–655 (1990).

    MathSciNet  MATH  Google Scholar 

  11. A. E. Mironov, “Self-adjoint commuting ordinary differential operators,” Invent. Math., 197, 417–431 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. A. E. Mironov, “Periodic and rapid decay rank two self-adjoint commuting differential operators,” in: Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014 (Amer. Math. Soc. Transl. Ser. 2, Vol. 234, V. M. Buchstaber, B. A. Dubrovin, and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2014), pp. 309–322.

    Google Scholar 

  13. V. S. Oganesyan, “Commuting differential operators of rank 2 with polynomial coefficients,” Funct. Anal. Appl., 50, 54–61 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Oganesyan, “Explicit characterization of some commuting differential operators of rank 2,” Int. Math. Res. Notices, 2017, 1623–1640 (2017).

    MathSciNet  Google Scholar 

  15. V. S. Oganesyan, “Common eigenfunctions of commuting differential operators of rank 2,” Math. Notes, 99, 308–311 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. N. Davletshina, “Commuting differential operators of rank 2 with trigonometric coefficients,” Siberian Math. J., 56, 405–410 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. O. I. Mokhov, “On commutative subalgebras of the Weyl algebra related to commuting operators of arbitrary rank and genus,” Math. Notes, 94, 298–300 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  18. O. I. Mokhov, “Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients,” in: Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014 (Amer. Math. Soc. Transl. Ser. 2, Vol. 234, V. M. Buchstaber, B. A. Dubrovin, and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2014), pp. 323–336.

    Google Scholar 

  19. A. E. Mironov and A. B. Zheglov, “Commuting ordinary differential operators with polynomial coefficients and automorphisms of the first Weyl algebra,” Int. Math. Res. Notices, 2016, 2974–2993 (2016).

    Article  MathSciNet  Google Scholar 

  20. V. Oganesyan, “Matrix commuting differential operators of rank 2,” Int. Math. Res. Notices, 2017, 1623–1640 (2017).

    MathSciNet  Google Scholar 

  21. A. O. Smirnov, “Finite-gap elliptic solutions of the KdV equation,” Acta Appl. Math., 36, 125–166 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. O. Smirnov, “Real finite-gap regular solutions of the Kaup–Boussinesq equation,” Theor. Math. Phys., 66, 19–31 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. O. Smirnov, “Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg–de Vries equation,” Russian Acad. Sci. Sb. Math., 82, 461–470 (1995).

    MathSciNet  Google Scholar 

  24. B. A. Dubrovin, “Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties,” Funct. Anal. Appl., 11, 265–277 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  25. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear,” Russ. Math. Surveys, 31, 59–146 (1976).

    Article  ADS  MATH  Google Scholar 

  26. F. Gesztesy and R. Weikard, “A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy,” Acta Math., 181, 63–108 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Weikard, “On commuting matrix differential operators,” New York J. Math., 8, 9–30 (2002).

    MathSciNet  MATH  Google Scholar 

  28. E. L. Ince, Ordinary differential equations, Dover, New York (1956).

    Google Scholar 

  29. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, Fla. (1984).

    MATH  Google Scholar 

  30. F. Gesztesy, K. Unterkofler, and R. Weikard, “An explicit characterization of Calogero–Moser systems,” Trans. Amer. Math. Soc., 358, 603–656 (2006).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    V. S. Oganesyan

Authors
  1. V. S. Oganesyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. S. Oganesyan.

Additional information

This research is supported by a grant from the Russian Science Foundation (Project No. 16-11-10260).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 1, pp. 50–63, July, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oganesyan, V.S. The AKNS Hierarchy and Finite-Gap Schrödinger Potentials. Theor Math Phys 196, 983–995 (2018). https://doi.org/10.1134/S004057791807005X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S004057791807005X

Keywords

Search

Navigation