Science China Mathematics

, Volume 53, Issue 5, pp 1195–1206 | Cite as

The determinant representation of the gauge transformation for the discrete KP hierarchy

  • ShaoWei Liu
  • Yi Cheng
  • JingSong HeEmail author


A successive gauge transformation operator T n+k for the discrete KP (dKP) hierarchy is defined, which is involved with two types of gauge transformations operators. The determinant representation of the T n+k is established and it is used to get a new τ function τ Δ (n+k) 4 of the dKP hierarchy from an initial τ Δ. In this process, we introduce a generalized discrete Wronskian determinant and some useful properties of discrete difference operators.


gauge transformation dKP hierarchy τ function 


35Q51 37K10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler V E, Bobenko A I, Suris Y B. Classification of integrable equation on quad-graphs: The consistency approach. Comm Math Phys, 2003, 233: 513–543zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aratyn H, Nissimov E, Pacheva S. Constrained KP hierarchies: Additional symmetries, Darboux-Bäcklund solutions and relations to multi-matrix models. Internat J Modern Phys A, 1997, 12: 1265–1340zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bobenko A I, Suris Y B. Integrable systems on quad-graphs. Int Math Res Not, 2002, 11: 573–611CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bobenko A I, Suris Y B. On discretization principles for differential geometry. Russian Math Surveys, 2007, 62: 1–43zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chau L L, Shaw J C, Yen H C. Solving the KP hierarchy by gauge transformations. Comm Math Phys, 1992, 149: 263–278zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chaw L L, Shaw J C, Tu M H. Solving the constrained KP hierarchy by gauge transformations. J Math Phys, 1997, 38: 4128–4137CrossRefMathSciNetGoogle Scholar
  7. 7.
    Date E, Kashiwara M, Jimbo M, et al. Transformation Groups for Soliton Equations. In: Nonlinear Integrable System: Classical Theory and Quantum Theory. Singapore: World Scientific, 1983, 39–119Google Scholar
  8. 8.
    Dickey L A. Modified KP and discrete KP. Lett Math Phys, 1999, 48: 277–289zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dickey L A. Soliton Equations and Hamiltonian Systems. Singapore: World Scientific, 2003zbMATHGoogle Scholar
  10. 10.
    Haine L, Iliev P. Commutative rings of difference operators and an adelic flag manifold. Int Math Res Not, 2000, 6: 281–323CrossRefMathSciNetGoogle Scholar
  11. 11.
    He J S, Cheng Y, Rudolf R. Solving bi-directional soliton equations in the KP hierarchy by gauge transformation. J High Energy Phys, 2006, 103: 1–38Google Scholar
  12. 12.
    He J S, Li Y S, Cheng Y. The determinant representation of the gauge transformation operators. Chinese Ann Math Ser B, 2002, 23: 475–486zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    He J S, Li Y S, Cheng Y. Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy. J Math Phys, 2003, 44: 3928–3960zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    He J S, Li Y S, Cheng Y. Q-deformed KP hierarchy and q-deformed constrained KP hierarchy. SIGMA, 2006, 2: 1–32MathSciNetGoogle Scholar
  15. 15.
    He J S, Wu Z W, Cheng Y. Gauge transformations for the constrained CKP and BKP hierarchies. J Math Phys, 2007, 48(113519): 1–16MathSciNetGoogle Scholar
  16. 16.
    Kupershmidt B A. Discrete Lax equations and differenece calculus. Astérisque, 1985, 123: 1–212Google Scholar
  17. 17.
    Oevel W. Darboux theorems and Wronskian formulas for integrable systems I: Constrained KP flows. Physica A, 1993, 195: 533–576zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Oevel W. Darboux transformations for integrable lattice systems. In: Nonlinear Physics: Theory and Experiment. Singapore: World Scientific, 1996, 233–240Google Scholar
  19. 19.
    Suris Y B. The problem of integrable discretization: Hamilton approach. In: Progress in Mathematics, vol. 219. Basel: Birkhäser-Verlag, 2003Google Scholar
  20. 20.
    Tu M H. Q-deformed KP hierarchy: Its additional symmetries and infinitesimal Bäcklund transformations. Lett Math Phys, 1999, 99: 95–103CrossRefGoogle Scholar
  21. 21.
    Tu M H, Shaw J C, Lee C R. On Darboux-Bäcklund transformation for the q-deformed Korteweg-de Vries hierarchy. Lett Math Phys, 1999, 49: 33–35zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHeFeiChina
  2. 2.Department of MathematicsNingbo UniversityNingboChina

Personalised recommendations