Skip to main content
SpringerLink
Log in

Additional symmetries of constrained CKP and BKP hierarchies

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction {Φ i , Ψ i } are presented explicitly. Furthermore, we show that acting on the space of the wave operator, ∂ k * forms new centerless W cC1=∞ and W cB1+∞ W cB1+∞ -subalgebra of centerless W 1+∞ respectively. In order to define above symmetry flows ∂ k * of the cCKP and cBKP hierarchies, two vital operators Y k are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.

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. Adler M, Shiota T, van Moerbke P. From the w ∞-algebra to its central extension: A τ-function approach. Phys Lett A, 1994, 194: 33–43

    Article  MathSciNet  Google Scholar 

  2. Adler M, Shiota T, van Moerbeke P. A Lax representation for the vertex operator and the central extension. Comm Math Phys, 1995, 171: 547–588

    Article  MATH  MathSciNet  Google Scholar 

  3. Alexandrov A, Mironov A, Morozov A. Solving Virasoro constraints in matrix models. Fortschr Phys, 2005, 53: 512–521

    Article  MATH  MathSciNet  Google Scholar 

  4. Aratyn H, Gomes J F, Zimerman A H. Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion. J Geom Phys, 2003, 46: 21–47

    Article  MATH  MathSciNet  Google Scholar 

  5. Aratyn H, Nissimov E, Pacheva S. Virasoro symmetry of constrained KP Hierarchies. Phys Lett A, 1997, 228: 164–175

    Article  MATH  MathSciNet  Google Scholar 

  6. Cheng Y. Constraints of the Kadomtsev-Petviashvili hierarchy. J Math Phys, 1992, 33: 3774–3782

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheng Y, Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys Lett A, 1991, 157: 22–26

    Article  MathSciNet  Google Scholar 

  8. Date E, Kashiwara M, Jimbo M, et al. KP hierarchy of orthogonal symplectic type—transformation groups for soliton equations VI. J Phys Soc Japan, 1981, 50: 3813–3818

    Article  MATH  MathSciNet  Google Scholar 

  9. Date E, Kashiwara M, Jimbo M, et al. Nonlinear Integrable Systems—Classical and Quantum Theory. Singapore: World Scientific, 1983, 39–119

    Google Scholar 

  10. Dickey L A. Additional symmetries of KP, Grassmannian, and the string equation. Mod Phys Lett A, 1993, 8: 1259–1272

    Article  MATH  MathSciNet  Google Scholar 

  11. Dickey L A. On additional symmetries of the KP hierarchy and Sato’s Bäcklund transformation. Comm Math Phys, 1995, 167: 227–233

    Article  MATH  MathSciNet  Google Scholar 

  12. Dickey L A. Soliton Equations and Hamiltonian Systems, 2nd ed. Singapore: World Scientific, 2003

    MATH  Google Scholar 

  13. He J S, Tian K L, Foerster A, et al. Additional symmetries and string equation of the CKP hierarchy. Lett Math Phys, 2007, 81: 119–134

    Article  MATH  MathSciNet  Google Scholar 

  14. He J S, Wu Z W, Cheng Y. Gauge transformations for the constrained CKP and BKP hierarchies. J Math Phys, 2007, 48: 113519

    Article  MathSciNet  Google Scholar 

  15. Konopelchenko B G, Sidorenko J, Strampp W. (1 + 1)-dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems. Phys Lett A, 1991, 157: 17–21

    Article  MathSciNet  Google Scholar 

  16. Loris I. On reduced CKP equations. Inverse Problems, 1999, 15: 1099–1109

    Article  MATH  MathSciNet  Google Scholar 

  17. Loris I. Symmetry reductions of the BKP hierarchy. J Math Phys, 1999, 40: 1420–1431

    Article  MATH  MathSciNet  Google Scholar 

  18. Loris I. Dimensional reductions of BKP and CKP hierarchies. J Phys A, 2001, 34: 3447–3459

    Article  MATH  MathSciNet  Google Scholar 

  19. Ma W X. K symmetries and τ symmetries of evolution equations and their Lie algebras. J Phys A, 1990, 23: 2707–2716

    Article  MATH  MathSciNet  Google Scholar 

  20. Ma W X. The algebraic structures of isospectral Lax operators and applications to integrable equations. J Phys A, 1992, 25: 5329–5343

    Article  MATH  MathSciNet  Google Scholar 

  21. Ma W X. Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations. J Math Phys, 1992, 33: 2464–2476

    Article  MATH  MathSciNet  Google Scholar 

  22. Mironov A. WDVV equations in Seiberg-Witten theory and associative algebras. Nuclear Phys B Proc Suppl, 1998, 61: 177–185

    Article  MathSciNet  Google Scholar 

  23. Mironov A, Morozov A. Virasoro constraints for Kontsevich-Hurwitz partition function. J High Energy Phys, 2009, doi: 10.1088/1126-6708/2009/02/024

  24. van Moerbke P. Integrable Foundation of String Theory. In: Lectures on Integrable Systems. Singapore: World Scientific, 1994, 163–267

    Google Scholar 

  25. Morozov A. Integrability and matrix models. Phys Usp, 1994, 37: 1–55

    Article  Google Scholar 

  26. Orlov A Yu, Schulman E I. Additional symmetries of integrable equations and conformal algebra reprensentaion. Lett Math Phys, 1986, 12: 171–179

    Article  MATH  MathSciNet  Google Scholar 

  27. Shen H F, Tu M H. On the string equation of the BKP hierarchy. Int J Mode Phys A, 2009, 24: 4193–4208

    Article  MATH  MathSciNet  Google Scholar 

  28. Takasaki K. Quasi-classical limit of BKP hierarchy and W-infity symmetries. Lett Math Phys, 1993, 28: 177–185

    Article  MATH  MathSciNet  Google Scholar 

  29. Tu M H. On the BKP hierarchy: Additional symmetries, Fay identity and Adler-Shiota-van Moerbeke formula. Lett Math Phys, 2007 81: 91–105

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    KeLei Tian, JiPeng Cheng & Yi Cheng

  2. Department of Mathematics, Ningbo University, Ningbo, 315211, China

    JingSong He

Authors
  1. KeLei Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JingSong He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JiPeng Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yi Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JingSong He.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tian, K., He, J., Cheng, J. et al. Additional symmetries of constrained CKP and BKP hierarchies. Sci. China Math. 54, 257–268 (2011). https://doi.org/10.1007/s11425-010-4076-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4076-6

Keywords

MSC(2000)

Search

Navigation