String equations of the q-KP hierarchy

  • Kelei Tian
  • Jingsong Hea
  • Yucai Su
  • Yi Cheng


Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators {L n , n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.


q-KP hierarchy Additional symmetry String equations Virasoro constraints 

2000 MR Subject Classification

35Q53 37K05 37K10 


  1. [1]
    Klimyk, A. and Schmüdgen, K., Quantum Groups and Their Representations, Springer-Verlag, Berlin, 1997.zbMATHGoogle Scholar
  2. [2]
    Kac, V. and Cheung, P., Quantum Calculus, Springer-Verlag, New York, 2002.zbMATHCrossRefGoogle Scholar
  3. [3]
    Zhang, D. H., Quantum deformation of KdV hierarchies and their infinitely many conservation laws, J. Phys. A, 26, 1993, 2389–2407.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Wu, Z. Y., Zhang, D. H. and Zheng, Q. R., Quantum deformation of KdV hierarchies and their exact solutions: q-deformed solitons, J. Phys. A, 27, 1994, 5307–5312.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Mas, J. and Seco, M., The algebra of q-pseudodifferential symbols and the q-W KP(n) algebra, J. Math. Phys., 37, 1996, 6510–6529.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Frenkel, E. and Reshetikhin, N., Quantum affine algebras and deformations of the Virasoro and W-algebras, Comm. Math. Phys., 178, 1996, 237–264.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Frenkel, E., Deformations of the KdV hierarchy and related soliton equations, Int. Math. Res. Not., 2, 1996, 55–76.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Khesin, B., Lyubashenko, V. and Roger, C., Extensions and contractions of the Lie algebra of qpseudodifferential symbols on the circle, J. Funct. Anal., 143, 1997, 55–97.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Haine, L. and Iliev, P., The bispectral property of a q-deformation of the Schur polynomials and the q-KdV hierarchy, J. Phys. A, 30, 1997, 7217–7227.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Iliev, P., Solutions to Frenkel’s deformation of the KP hierarchy, J. Phys. A, 31, 1998, 241–244.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Iliev, P., Tau function solutions to a q-deformation of the KP hierarchy, Lett. Math. Phys., 44, 1998, 187–200.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Tu, M. H., q-deformed KP hierarchy: its additional symmetries and infinitesimal Bäcklund transformations, Lett. Math. Phys., 49, 1999, 95–103.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Iliev, P., q-KP hierarchy, bispectrality and Calogero-Moser systems, J. Geom. Phys., 35, 2000, 157–182.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    He, J. S., Li, Y. H. and Cheng, Y., q-deformed KP hierarchy and its constrained sub-hierarchy, SIGMA, 2, 2006, 060, 33 pages.Google Scholar
  15. [15]
    Morozov, A., Integrability and matrix models, Phys. Usp, 37, 1994, 1–55.zbMATHCrossRefGoogle Scholar
  16. [16]
    Mironov, A., WDVV equations in Seiberg-Witten theory and associative algebras, Nuclear Phys. B Proc. Suppl., 61, 1998, 177–185.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Aratyn, H., Gomes, J. F. and Zimerman, A. H., Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion, J. Geom. Phys., 46, 2003, 21–47.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Alexandrov, A., Mironov, A. and Morozov, A., Solving Virasoro constraints in matrix models, Fortschr. Phys., 53, 2005, 512–521.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Mironov, A. and Morozov, A., Virasoro constraints for Kontsevich-Hurwitz partition function, J. High Energy Phys., 02, 2009, 024, 51 pages.Google Scholar
  20. [20]
    Shen, H. F. and Tu, M. H., On the string equation of the BKP hierarchy, Int. J. Mode. Phys. A, 24(22), 2009, 4193–4208.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Date, E., Kashiwara, M., Jimbo, M. and Miwa, T., Transformation groups for soliton equations, Nonlinear Integrable Systems-Classical and Quantum Theory, M. Jimbo and T. Miwa (eds.), World Scientific, Singapore, 1983, 39–119.Google Scholar
  22. [22]
    Orlov, A. Yu. and Schulman, E. I., Additional symmetries of integrable equations and conformal algebra reprensentaion, Lett. Math. Phys., 12, 1986, 171–179.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Dickey, L. A., On additional symmetries of the KP hierarchy and Sato’s Backlund transformation, Comm. Math. Phys., 167, 1995, 227–233.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Adler, M., Shiota, T. and van Moerbeke, P., A Lax representation for the vertex operator and the central extension, Comm. Math. Phys., 171, 1995, 547–588.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Adler, M., Shiota, T. and van Moerbeke, P., From the w -algebra to its central extension: a τ-function approach, Phys. Lett. A, 194, 1994, 33–43.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Panda, S. and Roy, S., The Lax operator approach for the Virasoro and the W-constraints in the generalized KdV hierarchy, Internat. J. Modern Phys. A, 8, 1993, 3457–3478.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Panda, S. and Roy, S., Remarks on the additional symmetries and W-constraints in the generalized KdV hierarchy, Phys. Lett. B, 296, 1992, 23–27.MathSciNetCrossRefGoogle Scholar
  28. [28]
    Dickey, L. A., Soliton Equations and Hamiltonian Systems, 2nd ed., World Scientific, Singapore, 2003.zbMATHGoogle Scholar
  29. [29]
    Dickey, L. A., Additional symmetries of KP, Grassmannian, and the string equation, Mod. Phys. Lett. A, 8, 1993, 1259–1272.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Wu Wen-Tsun Key Laboratory of Mathematics, Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of MathematicsNingbo UniversityNingboZhejiang, China
  3. 3.Department of MathematicsTongji UniversityShanghaiChina

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