Skip to main content

Advertisement

SpringerLink
Log in

Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Based on the characteristic polynomial of Lax matrix for the hierarchy of coupled Toda lattices associated with a \(3\times3\) discrete matrix spectral problem, we introduce a trigonal curve with two infinite points, from which we establish the associated Dubrovin-type equations. The asymptotic properties of the meromorphic function and the Baker-Akhiezer function are studied near two infinite points on the trigonal curve. Finite-band solutions of the entire hierarchy of coupled Toda lattices are obtained in terms of the Riemann theta function.

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. Krichever, I.M.: Algebraic-geometric construction of the Zaharov-Sabat equations and their periodic solutions. Sov. Math. Dokl. 17, 394–397 (1976)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Dubrovin, B.A.: Theta functions and nonlinear equations. Russ. Math. Surv. 36, 11–92 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubrovin, B.A.: Matrix finite-gap operators. Rev. Sci. Tech. 23, 33–78 (1983)

    MathSciNet  Google Scholar 

  5. Belokolos, E.D., Bobenko, A.I., Enolskij, V.Z., Its, A.R., Mateveev, V.B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)

    Google Scholar 

  6. Date, E., Tanaka, S.: Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice. Prog. Theor. Phys. Suppl. 59, 107–125 (1976)

    Article  MathSciNet  Google Scholar 

  7. Ma, Y.C., Ablowitz, M.J.: The periodic cubic Schrödinger equation. Stud. Appl. Math. 65, 113–158 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lia, L.C., Nenciu, I.: The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices. Adv. Math. 231, 3330–3388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Previato, E.: Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schroedinger equation. Duke Math. J. 52, 329–377 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Miller, P.D., Ercolani, N.N., Krichever, I.M., Levermore, C.D.: Finite genus solutions to the Ablowitz-Ladik equations. Commun. Pure Appl. Math. 48, 1369–1440 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gesztesy, F., Ratneseelan, R.: An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10, 345–391 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Alber, M.S., Fedorov, Y.N.: Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Probl. 17, 1017–1042 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geronimo, J.S., Gesztesy, F., Holden, H.: Algebro-geometric solutions of the Baxter-Szegö difference equation. Commun. Math. Phys. 258, 149–177 (2005)

    Article  MATH  Google Scholar 

  15. Gesztesy, F., Holden, H., Michor, J., Teschl, G.: Soliton Equations and Their Algebro-geometric Solutions, Vol. II. (\(1+1\))-Dimensional Discrete Models. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  16. Cao, C.W., Wu, Y.T., Geng, X.G.: Relation between the Kadomtsev-Petviashvili equation and the confocal involutive system. J. Math. Phys. 40, 3948–3970 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Geng, X.G., Cao, C.W.: Decomposition of the (\(2+1\))-dimensional Gardner equation and its quasi-periodic solutions. Nonlinearity 14, 1433–1452 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Geng, X.G., Dai, H.H., Zhu, J.Y.: Decomposition of the discrete Ablowitz-Ladik hierarchy. Stud. Appl. Math. 118, 281–312 (2007)

    Article  MathSciNet  Google Scholar 

  19. Buchstaber, V.M., Enolskii, V.Z., Leykin, D.V.: Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations. Funct. Anal. Appl. 34, 159–171 (2000)

    Article  MathSciNet  Google Scholar 

  20. Brezhnev, Y.V.: Finite-band potentials with trigonal curves. Theor. Math. Phys. 133, 1657–1662 (2002)

    Article  MATH  Google Scholar 

  21. Baldwin, S., Eilbeck, J.C., Gibbons, J., Ônishi, Y.: Abelian functions for cyclic trigonal curves of genus 4. J. Geom. Phys. 58, 450–467 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eilbeck, J.C., Enolski, V.Z., Matsutani, S., Ônishi, Y., Previato, E.: Abelian functions for purely trigonal curves of genus three. Int. Math. Res. Not. 2007, rnm140 (2007), 38 pp.

    MATH  Google Scholar 

  23. Ônishi, Y.: Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case. Int. J. Math. 20, 427–441 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Matveev, V.B., Smirnov, A.O.: On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations. Lett. Math. Phys. 14, 25–31 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Matveev, V.B., Smirnov, A.O.: Symmetric reductions of the Riemann-function and some of their applications to the Schrödinger and Boussinesq equations. Transl. Am. Math. Soc. 157, 227–237 (1993)

    MATH  Google Scholar 

  26. Previato, E.: The Calogero-Moser-Krichever system and elliptic Boussinesq solitons. In: Harnad, J., Marsden, J.E. (eds.) Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, pp. 57–67. CRM, Monreal (1990)

    Google Scholar 

  27. Previato, E.: Monodromy of Boussinesq elliptic operators. Acta Appl. Math. 36, 49–55 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Previato, E., Verdier, J.L.: Boussinesq elliptic solitons: the cyclic case. In: Ramanan, S., Beauville, A. (eds.) Proc. Indo-French Conf. on Geometry, Delhi, pp. 173–185. Hindustan Book Agency, Delhi (1993)

    Google Scholar 

  29. Smirnov, A.O.: A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann theta-functions. Math. USSR Sb. 61, 379–388 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation a related many-body problem. Commun. Pure Appl. Math. 30, 95–148 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  31. McKean, H.P.: Integrable systems and algebraic curves. In: Grmela, M., Marsden, J.E. (eds.) Global Analysis. Lecture Notes in Math., vol. 755, pp. 83–200. Springer, Berlin (1979)

    Chapter  Google Scholar 

  32. Dickson, R., Gesztesy, F., Unterkofler, K.: A new approach to the Boussinesq hierarchy. Math. Nachr. 198, 51–108 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dickson, R., Gesztesy, F., Unterkofler, K.: Algebro-geometric solutions of the Boussinesq hierarchy. Rev. Math. Phys. 11, 823–879 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Geng, X.G., Wu, L.H., He, G.L.: Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions. Physica D 240, 1262–1288 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Geng, X.G., Wu, L.H., He, G.L.: Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy. J. Nonlinear Sci. 23, 527–555 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Geng, X.G., Zhai, Y.Y., Dai, H.H.: Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy. Adv. Math. 263, 123–153 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wu, L.H., Geng, X.G., Zhang, J.S.: Algebro-geometric solution to the Bullough-Dodd-Zhiber-Shabat equation. Int. Math. Res. Not. 2015, 2141–2167 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Blaszak, M., Marciniak, K.: R-matrix approach to lattice integrable systems. J. Math. Phys. 35, 4661–4682 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hisakado, M.: Hungry Volterra equation, multi Boson KP hierarchy and two matrix models. J. Phys. Soc. Jpn. 68, 2221–2227 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, Y.T., Geng, X.G.: A new integrable symplectic map of Neumann type. J. Phys. Soc. Jpn. 68, 784–790 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhu, Z.N., Wu, X.N., Xue, W.M., Ding, Q.: Infinitely many conservation laws of two Blaszak-Marciniak three-field lattice hierarchies. Phys. Lett. A 297, 387–395 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu, Y.T., Geng, X.G.: A finite-dimensional integrable system associated with the three-wave interaction equations. J. Math. Phys. 40, 3409–3430 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)

    Book  MATH  Google Scholar 

  44. Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant nos. 11331008 and 11601488).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, People’s Republic of China

    Xin Zeng & Xianguo Geng

Authors
  1. Xin Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianguo Geng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianguo Geng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeng, X., Geng, X. Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices. Acta Appl Math 154, 59–81 (2018). https://doi.org/10.1007/s10440-017-0133-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-017-0133-2

Keywords

Search

Navigation