Theoretical and Mathematical Physics

, Volume 179, Issue 3, pp 649–678 | Cite as

Quasiperiodic solutions of the discrete Chen-Lee-Liu hierarchy

  • Xin Zeng
  • Xianguo GengEmail author


Using the Lax matrix and elliptic variables, we decompose the discrete Chen-Lee-Liu hierarchy into solvable ordinary differential equations. Based on the theory of the algebraic curve, we straighten the continuous and discrete flows related to the discrete Chen-Lee-Liu hierarchy in Abel-Jacobi coordinates. We introduce the meromorphic function ϕ, Baker-Akhiezer vector \(\bar \psi \), and hyperelliptic curve ɛN according to whose asymptotic properties and the algebro-geometric characters we construct quasiperiodic solutions of the discrete Chen-Lee-Liu hierarchy.


discrete Chen-Lee-Liu equation quasiperiodic solution 


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  1. 1.
    R. Hirota, J. Phys. Soc. Japan, 35, 289–294 (1973).ADSCrossRefGoogle Scholar
  2. 2.
    H. Flaschka, Prog. Theoret. Phys., 51, 703–716 (1974).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. J. Ablowitz and J. F. Ladik, J. Math. Phys., 16, 598–603 (1975).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    M. J. Ablowitz and J. F. Ladik, J. Math. Phys., 17, 1011–1018 (1975).ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math., 55, 213–229 (1976).zbMATHMathSciNetGoogle Scholar
  6. 6.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Inverse Problem Method [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Method of Inverse Scattering, Consultants Bureau (1984).zbMATHGoogle Scholar
  7. 7.
    T. Tsuchida, J. Phys. A, 35, 7827–7847 (2002); arXiv:nlin/0105053v3 (2001).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    H. H. Chen, Y. C. Lee, and C. S. Liu, Phys. Scripta, 20, 490–492 (1979).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    E. Date and S. Tanaka, Prog. Theoret. Phys., 55, 457–465 (1976).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    N. N. Bogolyubov Jr., A. K. Prikarpatskii, and V. G. Samoilenko, Theor. Math. Phys., 50, 75–81 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    N. N. Bogolyubov and A. K. Prikarpatskii, Soviet. Phys. Dokl., 27, 113–116 (1982).ADSzbMATHGoogle Scholar
  12. 12.
    S. Ahmad and A. R. Chowdhury, J. Math. Phys., 28, 134–137 (1987).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    S. J. Alber, Lett. Math. Phys., 17, 149–155 (1989).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    P. D. Miller, N. M. Ercolani, I. M. Krichever, and C. D. Levermore, Commun. Pure Appl. Math., 48, 1369–1440 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    V. E. Vekslerchik, J. Phys. A, 32, 4983–4994 (1999).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Mem. Amer. Math. Soc., 135, 1–79 (1998).MathSciNetGoogle Scholar
  17. 17.
    X. G. Geng, H. H. Dai, and C. W. Cao, J. Math. Phys., 44, 4573–4588 (2003).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    J. S. Geronimo, F. Gesztesy, and H. Holden, Commun. Math. Phys., 258, 149–177 (2005).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    C. W. Cao, X. G. Geng, and H. Y. Wang, J. Math. Phys., 43, 621–643 (2002).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    X. G. Geng, H. H. Dai, and J. Y. Zhu, Stud. Appl. Math., 118, 281–312 (2007).MathSciNetGoogle Scholar
  21. 21.
    I. M. Krichever, Sov. Math. Dokl., 17, 394–397 (1976).zbMATHGoogle Scholar
  22. 22.
    B. A. Dubrovin, Funct. Anal. Appl., 9, 215–223 (1975).CrossRefGoogle Scholar
  23. 23.
    B. A. Dubrovin, Russ. Math. Surveys, 36, 11–92 (1981).ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).zbMATHGoogle Scholar
  25. 25.
    E. Date and S. Tanaka, Progr. Theoret. Phys. Suppl., 59, 107–125 (1976).ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Y. C. Ma and J. M. Ablowitz, Stud. Appl. Math., 65, 113–158 (1981).zbMATHMathSciNetGoogle Scholar
  27. 27.
    F. Gesztesy and R. Ratneseelan, Rev. Math. Phys., 10, 345–391 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    M. S. Alber and Y. N. Fedorov, Inverse Problems, 17, 1017–1042 (2001).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    C. W. Cao, Y. T. Wu, and X. G. Geng, J. Math. Phys., 40, 3948–3970 (1999).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    X. G. Geng and C. W. Cao, Nonlinearity, 14, 1433–1452 (2001).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1994).CrossRefzbMATHGoogle Scholar
  32. 32.
    D. Mumford, Tata Lectures on Theta (Prog. Math., Vol. 43), Vol. II, Jacobian Theta Functions and Differential Equations, Birkhäuser, Boston (1984).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhou, HenanChina

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