Skip to main content
SpringerLink
Log in

The gauge transformations of the constrained q-deformed modified KP hierarchy and their relations with the additional symmetries

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, two kinds of gauge transformations for the constrained q-deformed modified KP hierarchy and their relations with additional symmetries are discussed. The corresponding gauge transformations are required to keep both the Lax equation and the Lax operator, which can be obtained by choosing the special generating functions. Then the successive gauge transformations for the constrained q-modified KP hierarchy are investigated. Finally, we find that additional symmetry flows commute with the gauge transformation for the q-mKP hierarchy.

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. Kac, V., Cheung, P.: Quantum Calculus. Springer-Verlag, New York (2002)

    Book  Google Scholar 

  2. Iliev, P.: Tau function solutions to a \(q\)-deformation of the KP hierarchy. Lett. Math. Phys. 44, 187–200 (1998)

    Article  MathSciNet  Google Scholar 

  3. He, J.S., Li, Y.H., Cheng, Y.: \(q\)-deformed KP hierarchy and \(q\)-deformed constrained KP hierarchy. SIGMA 2, 060 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Tian, K.L., Zhu, X.M., He, J.S.: On recursion operator of the \(q\)-KP hierarchy. Commun. Theor. Phys. 66, 263–268 (2016)

    Article  MathSciNet  Google Scholar 

  5. Tian, K.L., He, J.S., Su, Y.C.: Symmetric \(q\)-deformed KP hierarchy. Chin. Ann. Math. 36, 1–10 (2015)

    Article  MathSciNet  Google Scholar 

  6. Tu, M.H., Shaw, J.C., Lee, H.C.: On Darboux–Backlund transformations for the \(q\)-deformed Korteweg–de Vries hierarchy. Lett. Math. Phys. 49, 33–45 (1999)

    Article  MathSciNet  Google Scholar 

  7. Takasaki, K.: \(q\)-analogue of modified KP hierarchy and its quasi-classical limit. Lett. Math. Phys. 72, 165–181 (2005)

    Article  MathSciNet  Google Scholar 

  8. Cheng, J.P.: Miura and auto-Backlund transformations for the \(q\)-deformed KP and \(q\)-deformed modified KP hierarchies. J. Nonlinear Math. Phys. 24, 7–19 (2017)

    Article  MathSciNet  Google Scholar 

  9. Tu, M.H., Shaw, J.C., Lee, C.R.: On the \(q\)-deformed modified Korteweg–de Vries hierarchy. Phys. Lett. A 266, 155–159 (2000)

    Article  MathSciNet  Google Scholar 

  10. Tian, K.L., Ge, Y.Y., Zhu, X.M.: On the \(q\)-deformed modified Kadomtsev–Petviashvili hierarchy and its additional simmetries. Rom. Rep. Phys. 69, 110 (2017)

    Google Scholar 

  11. Tian, K.L., Ge, Y.Y., Zhu, X.M.: The q-deformed mKP hierarchy with two parameters. Mod. Phys. Lett. B. 32, 1850170 (2018)

    Article  MathSciNet  Google Scholar 

  12. Lin, R.L., Peng, H., Manas, M.: The \(q\)-deformed mKP hierarchy with self-consistent sources, Wronskian soiutions and solitons. J. Phys. A: Math. Theor. 43, 434022 (2010)

    Article  Google Scholar 

  13. Cheng, J.P., Wang, J.Z., Zhang, X.Y.: The gauge transformation of the \(q\)-deformed modified KP hierarchy. J. Nonlin. Math. Phys. 21, 533–542 (2014)

    Article  MathSciNet  Google Scholar 

  14. Li, N., Cheng, J.P.: The successive applications of two types of gauge transformations for the \(q\)-deformed modified Kadomtsev–Petvivashvili hierarchy. Z. Naturforsch. A. 73, 345–356 (2018)

    Article  Google Scholar 

  15. Geng, L.M., Chen, H.Z., Li, N., Cheng, J.P.: The gauge transformations of the constrained \(q\)-deformed KP hierarchy. Mod. Phys. Lett. B. 32, 1850176 (2018)

    Article  MathSciNet  Google Scholar 

  16. Tian, K.L., He, J.S., Su, Y.C., Cheng, Y.: String equations of the q-KP hierarchy. Chin. Ann. Math. Ser. B 32, 895–904 (2011)

    Article  MathSciNet  Google Scholar 

  17. Tu, M.H.: \(q\)-deformed KP hierarchy: its additional symmetries and infinitesimal Backlund transformations. Lett. Math. Phys. 49, 95–103 (1999)

    Article  MathSciNet  Google Scholar 

  18. Lin, R.L., Liu, X.J., Zeng, Y.B.: A new extended \(q\)-deformed KP hierarchy. J. Nonlinear Math. Phys. 15, 333–347 (2008)

    Article  MathSciNet  Google Scholar 

  19. Li, C.Z.: Sato theory on the \(q\)-Toda hierarchy and its extension. Chaos Solitons Fractals 76, 10–23 (2015)

    Article  MathSciNet  Google Scholar 

  20. Wang, S.K., Wu, K., Wu, X.N., Yu, D.L.: The \(q\)-deformation of AKNS-D hierarchy. J. Phys. A: Math. Gen. 34, 9641–9651 (2001)

    Article  MathSciNet  Google Scholar 

  21. Orlov, A. Yu.: Volterra operator algebra for zero curvature representation. Universality of KP. In: Nonlinear Processes in Physics, pp. 126-131. Springer, Berlin, Heidelberg (1993)

  22. Orlov, A.Yu., Rauch-Wojciechowski, S.: Dressing method, Darboux transformation and generalized restricted flows for the KdV hierarchy. Phys. D 69, 77–84 (1993)

    Article  MathSciNet  Google Scholar 

  23. Enriquez, B., Orlov, A.Y., Rubtsov, V.N.: Dispersionful analogues of Benney’s equations and \(N\)-wave systems. Inverse Probl. 12, 241–250 (1996)

    Article  MathSciNet  Google Scholar 

  24. Oevel, W.: Darboux theorems and Wronskian formulas for integrable system I: constrained KP flows. Phys. A. 195, 533–576 (1993)

    Article  MathSciNet  Google Scholar 

  25. Oevel, W., Rogers, C.: Gauge transformations and reciprocal links in \(2+1\) dimensions. Rev. Math. Phys. 5, 299–330 (1993)

    Article  MathSciNet  Google Scholar 

  26. Chau, L.L., Shaw, J.C., Yen, H.C.: Solving the KP hierarchy by gauge transformations. Commun. Math. Phys. 149, 263–278 (1992)

    Article  MathSciNet  Google Scholar 

  27. He, J.S., Li, Y.S., Cheng, Y.: The determinant representation of the gauge transformation operators. Chin. Ann. Math. B 23, 475–486 (2002)

    Article  MathSciNet  Google Scholar 

  28. Aratyn, H., Nissimov, E., Pacheva, S.: Darboux-Backlund solutions of SL(p, q) KP-KdV hierarchies, constrained generalized Toda lattices, and two-matrix string model. Phys. Lett. A. 201, 293–305 (1995)

    Article  MathSciNet  Google Scholar 

  29. Chau, L.L., Shaw, J.C., Tu, M.H.: Solving the constrained KP hierarchy by gauge transformations. J. Math. Phys. 38, 4128–4137 (1997)

    Article  MathSciNet  Google Scholar 

  30. Orlov, A.Y., Schulman, E.I.: Additional symmetries for integrable equations and conformal algebra repesentation. Lett. Math. Phys. 12, 171–179 (1986)

    Article  MathSciNet  Google Scholar 

  31. Adler, M., Shiota, T., van Moerbeke, P.: From the \(w_{\infty }\)-algebra to its central extension: a \(\tau \)-function approach. Phys. Lett. A. 194, 33–43 (1994)

    Article  MathSciNet  Google Scholar 

  32. van Moerbeke, P.: Integrable fundations of string theory. In: Babelon, O., et al. (eds.) Lectures on Integrable systems, pp. 163–267. World Scientific, Singapore (1994)

    Google Scholar 

  33. Dickey, L.A.: On additional symmetries of the KP hierarchy and Sato’s Backlund transformation. Commun. Math. Phys. 167, 227–233 (1995)

    Article  Google Scholar 

  34. Orlov, AYu.: Vertex operator, \({\bar{\partial }}\)-problem, symmetries, Hamiltonian and Lagrangian formalism of (2+1) dimensional integrable systems. In: Baryakhtar, V., et al. (eds.) Plasma Theory and Nonlinear and Turbulent Processes in Physics, pp. 116–134. World Scientific, Singapore (1988)

    Google Scholar 

  35. Orlov, A.Yu., Winternitz, P.: Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev–Petviashvili hierarchy. J. Math. Phys 38, 4644–4674 (1997)

    Article  MathSciNet  Google Scholar 

  36. Grinevich, P.G., Orlov, AYu.: Virasoro action on Riemann surfaces, grassmannians, det \({\bar{\partial }}_{J}\) and Segal-Wilson \(\tau \)-Function. In: Belavin, A., et al. (eds.) Problems of Modern Quantum Field Theory, pp. 86–106. Springer, Berlin (1989)

    Chapter  Google Scholar 

  37. Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix modelsof two-dimensional gravity and Toda theory. Nuclear Phys. B 357, 565–618 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, People’s Republic of China

    Huizhan Chen

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Lumin Geng, Na Li & Jipeng Cheng

Authors
  1. Huizhan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lumin Geng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Na Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jipeng Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jipeng Cheng.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, H., Geng, L., Li, N. et al. The gauge transformations of the constrained q-deformed modified KP hierarchy and their relations with the additional symmetries. Anal.Math.Phys. 10, 79 (2020). https://doi.org/10.1007/s13324-020-00431-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00431-x

Keywords

Mathematics Subject Classification

Search

Navigation