Skip to main content
SpringerLink
Log in

The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker-Akhiezer Function

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

Abstract

We obtain a simple sufficient condition for the solvability of the Riemann factorization problem for matrix-valued functions on a circle. This condition is based on the symmetry principle. As an application, we consider nonlinear evolution equations that can be obtained by a unitary reduction from the zero-curvature equations connecting a linear function of the spectral parameter z and a polynomial of z. We consider solutions obtained by dressing the zero solution with a function holomorphic at infinity. We show that all such solutions are meromorphic functions on ℂ 2xt without singularities on ℝ 2xt . This class of solutions contains all generic finite-gap solutions and many rapidly decreasing solutions but is not exhausted by them. Any solution of this class, regarded as a function of x for almost every fixed t ∈ ℂ, is a potential with a convergent Baker-Akhiezer function for the corresponding matrix-valued differential operator of the first order.

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. N. I. Muskhelishvili, Singular Integral Equations [in Russian] (3rd ed.), Nauka, Moscow (1968); English transl. prev. ed., Nordhoff, Groningen (1961); N. P. Vekua, Systems of Singular Integral Equations and Some Boundary Problems [in Russian] (2nd ed.), Nauka, Moscow (1970); English transl.: Systems of Singular Integral Equations, Noordhoff, Groningen (1967).

    Google Scholar 

  2. S. Prossdorf, Some Classes of Singular Equations, North-Holland, Amsterdam (1978).

    Google Scholar 

  3. I. Ts. Gokhberg and M. G. Krein, Usp. Mat. Nauk, 13, No.2, 3–72 (1958); English transl.: Amer. Math. Soc. Transl., Ser. 2, 14 217–287 (1960).

    Google Scholar 

  4. A. N. Parshin, “To Chapter X: Riemann problem in the theory of functions of a complex variable [in Russian],” in: D. Hilbert: Selected Works, Vol. 2, Faktorial, Moscow (1998), pp. 535–538.

    Google Scholar 

  5. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 13, 166–174 (1980).

    Google Scholar 

  6. 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 transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Plenum, New York (1984).

    Google Scholar 

  7. L. A. Takhtadzhyan and L. D. Faddeev, The Hamiltonian Methods in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtajan, Berlin, Springer (1987).

    Google Scholar 

  8. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, Integrable Systems: Group Theory Approach [in Russian], Inst. Computer Studies, Moscow (2003).

    Google Scholar 

  9. Yu. L. Rodin, Phys. D, 24, 1–53 (1987); A. R. Its, Notices Am. Math. Soc., 50 1389–1400 (2003).

    Google Scholar 

  10. M. Sato, RIMS Kokyuroku, 439, 30–46 (1981).

    Google Scholar 

  11. G. B. Segal and G. Wilson, Publ. Math. IHES, 61, 5–65 (1985).

    Google Scholar 

  12. G. Haak, M. Schmidt, and R. Schrader, Rev. Math. Phys., 4, 451–499 (1992).

    Article  Google Scholar 

  13. G. Wilson, “The τ-functions of the \(\mathfrak{g}\)AKNS equations,” in: Integrable Systems: The Verdier Memorial Conf., Luminy, France, July 1–5, 1991 (Progr. Math., Vol. 115, O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach, eds.), Birkhauser, Basel (1993), pp. 131–145.

    Google Scholar 

  14. D. H. Sattinger and J. S. Szmigielski, Phys. D, 64, 1–34 (1993).

    Google Scholar 

  15. C.-L. Terng and K. Uhlenbeck, Comm. Pure Appl. Math., 53, 1–75 (2000).

    Article  Google Scholar 

  16. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems: I [in Russian],” in: Sovrem. Probl. Mat. Fund. Naprav., Vol. 4, Dynamical Systems —4 (V. I. Arnol’d and S. P. Novikov, eds.), VINITI, Moscow (1985), pp. 179–285; English transl.: “Integrable systems: I,” in: Dynamical Systems IV: Symplectic Geometry and its Applications (Encycl. Math. Sci., Vol. 4, V. I. Arnol’d, S. P. Novikov, and R. V. Gamkrelidze, eds.), Springer, Berlin (1990), pp. 173–280.

    Google Scholar 

  17. I. M. Krichever, Sov. Math. Dokl., 22, 79–84 (1980).

    Google Scholar 

  18. I. M. Krichever, J. Sov. Math., 28, 51–90 (1985).

    Article  Google Scholar 

  19. I. V. Cherednik, Sov. Phys. Dokl., 27, 716–718 (1982).

    Google Scholar 

  20. S. P. Novikov, Funct. Anal. Appl., 8, 236–246 (1974).

    Article  Google Scholar 

  21. A. G. Reiman and M. A. Semenov-Tian-Shansky, Invent. Math., 63, 423–432 (1981).

    Article  Google Scholar 

  22. F. Gesztesy, W. Karwowski, and Z. Zhao, Duke Math. J., 68, 101–150 (1992).

    Article  Google Scholar 

  23. A. Degasperis and A. Shabat, Theor. Math. Phys., 100, 970–984 (1994).

    Article  Google Scholar 

  24. V. Yu. Novokshenov, Phys. D, 87, 109–114 (1995).

    Google Scholar 

  25. V. E. Zakharov and A. B. Shabat, JETP, 34, 62–69 (1972).

    Google Scholar 

  26. A. B. Shabat, Differential Equations, 15, 1299–1307 (1980).

    Google Scholar 

  27. X. Zhou, “Zakharov-Shabat inverse scattering,” in: Scattering (R. Pike and P. Sabatier, eds.), Acad. Press, San Diego (2002), pp. 1707–1716.

    Google Scholar 

  28. I. M. Krichever, Funct. Anal. Appl., 11, 12–26 (1977).

    Article  Google Scholar 

  29. Yu. Laiterer, “Holomorphic vector bundles and the Oka-Grauert principle [in Russian],” in: Sovrem. Probl. Mat. Fund. Naprav., Vol. 10, Complex Analysis: Several Variables 4 (S. G. Gindikin and G. M. Khenkin, eds.), VINITI, Moscow (1986), pp. 75–121; English transl.: J. Leiterer, in: Several Complex Variables IV: Algebraic Aspects of Complex Analysis (Encycl. Math. Sci., Vol. 10, S. G. Gindikin, G. M. Khenkin, and R. V. Gamkrelidze, eds.), Springer, Berlin (1990), pp. 63–103.

    Google Scholar 

  30. K. Uhlenbeck, J. Differential Geom., 30, 1–50 (1989).

    Google Scholar 

  31. A. Pressley and G. Segal, Loop Groups, Clarendon, Oxford (1986); Russian transl., Mir, Moscow (1990).

    Google Scholar 

  32. F. Musso and A. B. Shabat, Theor. Math. Phys., 144, 1004–1013 (2005); nlin. SI/0502006 (2005).

    Article  Google Scholar 

  33. A. V. Domrin, Dokl. Math., 67, 349–351 (2003).

    Google Scholar 

  34. B. A. Dubrovin, J. Sov. Math., 28, 20–50 (1985).

    Article  Google Scholar 

  35. D. Wu, Inverse Problems, 18, 95–109 (2002).

    Article  Google Scholar 

  36. I. McIntosh, Nonlinearity, 7, 85–108 (1994).

    Article  Google Scholar 

  37. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966).

    Google Scholar 

  38. L. A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore (1991).

    Google Scholar 

  39. B. V. Shabat, Introduction to Complex Analysis [in Russian], Vol. 2, Nauka, Moscow (1985); English transl. Introduction to Complex Analysis: Part II. Functions of Several Variables (Transl. Math. Monogr., Vol. 110), Amer. Math. Soc., Providence, R. I. (1992).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. V. Domrin

Authors
  1. A. V. Domrin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 453–471, September, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Domrin, A.V. The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker-Akhiezer Function. Theor Math Phys 144, 1264–1278 (2005). https://doi.org/10.1007/s11232-005-0158-y

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-005-0158-y

Keywords

Search

Navigation