Skip to main content
SpringerLink
Log in

Darboux Transformations for the Supersymmetric Two-Boson Hierarchy

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we construct Darboux transformations and Bäcklund transformations for the supersymmetric two-boson hierarchy. Three Darboux transformations and associated Bäcklund transformations are obtained. For one of them, we also work out the corresponding nonlinear superposition formula. We build, for both the supersymmetric two-boson equation and the \(N=2\), \(a=4\) supersymmetric KdV equation, their one-soliton solutions and present their semi-discrete versions.

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. Kaup, D.J.: Finding eigenvalue problems for solving nonlinear evolution equations. Prog. Theor. Phys. 54, 72–78 (1975)

    Article  MathSciNet  Google Scholar 

  2. Kaup, D.J.: A higher-order water-wave equation and the method for solving it. Prog. Theor. Phys. 54, 396–408 (1975)

    Article  MathSciNet  Google Scholar 

  3. Broer, L.J.F.: Approximate equations for long water waves. Appl. Sci. Res. 31, 377–395 (1975)

    Article  MathSciNet  Google Scholar 

  4. Hirota, R., Satsuma, J.: Nonlinear evolution equations generated from the Bäcklund transformation for the Boussinesq equation. Prog. Theor. Phys. 57, 797–807 (1977)

    Article  Google Scholar 

  5. Nakamura, A., Hirota, R.: A new example of explode-decay solitary waves in one-dimension. J. Phys. Soc. Jpn. 54, 491–499 (1985)

    Article  MathSciNet  Google Scholar 

  6. Liu, Q.M., Hu, X.B., Li, Y.: Rational solutions of the classical Boussinesq hierarchy. J. Phys. A, Math. Gen. 23, 585–591 (1990)

    Article  MathSciNet  Google Scholar 

  7. Martínez Alonso, L., Medina Reus, E.: Soliton interaction with change of form in the classical Boussinesq system. Phys. Lett. A 167, 370–376 (1992)

    Article  MathSciNet  Google Scholar 

  8. Satsuma, J., Kajiwara, K., Matsukidaira, J., Hietarinta, J.: Solutions of the Broer-Kaup system through its trilinear form. J. Phys. Soc. Jpn. 61, 3096–3102 (1992)

    Article  Google Scholar 

  9. Kupershmidt, B.A.: Mathematics of dispersive water waves. Commun. Math. Phys. 99, 51–73 (1985)

    Article  MathSciNet  Google Scholar 

  10. Leo, R.A., Mancarella, G., Soliani, G.: On the Broer-Kaup hydrodynamical system. J. Phys. Soc. Jpn. 57, 753–756 (1988)

    Article  MathSciNet  Google Scholar 

  11. Gordoa, P.R., Conde, J.M.: A linear algebraic nonlinear superposition formula. Phys. Lett. A 295, 287–298 (2002)

    Article  MathSciNet  Google Scholar 

  12. Leble, S.B., Ustinov, N.V.: Korteweg-de Vries-modified Korteweg-de Vries systems and Darboux transforms in 1+1 and 2+1 dimensions. J. Math. Phys. 34, 1421–1428 (1993)

    Article  MathSciNet  Google Scholar 

  13. Li, Y., Ma, W.X., Zhang, J.E.: Darboux transformations of classical Boussinesq system and its new solutions. Phys. Lett. A 275, 60–66 (2000)

    Article  MathSciNet  Google Scholar 

  14. Li, Y., Zhang, J.E.: Darboux transformations of classical Boussinesq system and its multi-soliton solutions. Phys. Lett. A 284, 253–258 (2001)

    Article  MathSciNet  Google Scholar 

  15. Aratyn, H., Ferreira, L.A., Gomes, J.F., Zimerman, A.H.: On two-current realization of KP hierarchy. Nucl. Phys. B 402, 85–117 (1993)

    Article  MathSciNet  Google Scholar 

  16. Bonora, L., Xiong, C.S.: Matrix models without scaling limit. Int. J. Mod. Phys. A 8, 2973–2992 (1993)

    Article  MathSciNet  Google Scholar 

  17. Huang, W., Xue, L., Liu, Q.P.: Integrable discretizations for classical Boussinesq system. J. Phys. A, Math. Theor. 54, 045201 (2021)

    Article  MathSciNet  Google Scholar 

  18. Brunelli, J.C., Das, A.: The supersymmetric two boson hierarchies. Phys. Lett. B 337, 303–307 (1994)

    Article  MathSciNet  Google Scholar 

  19. Brunelli, J.C., Das, A.: Properties of nonlocal charges in the supersymmetric two boson hierarchy. Phys. Lett. B 354, 307–314 (1995)

    Article  MathSciNet  Google Scholar 

  20. Brunelli, J.C., Das, A.: Supersymmetric two-boson equation, its reductions and the nonstandard supersymmetric KP hierarchy. Int. J. Mod. Phys. A 10, 4563–4599 (1995)

    Article  MathSciNet  Google Scholar 

  21. Liu, Q.P., Yang, X.X.: Supersymmetric two-boson equation: bilinearization and solutions. Phys. Lett. A 351, 131–135 (2006)

    Article  MathSciNet  Google Scholar 

  22. Fan, E.: New bilinear Bäcklund transformation and Lax pair for the supersymmetric two-boson equation. Stud. Appl. Math. 127, 284–301 (2011)

    Article  MathSciNet  Google Scholar 

  23. Fan, E., Hon, Y.C.: Super extension of Bell polynomials with applications to supersymmetric equations. J. Math. Phys. 53, 013503 (2012)

    Article  MathSciNet  Google Scholar 

  24. Xue, L., Liu, Q.P.: On Bäcklund transformation for supersymmetric two-boson equation. Phys. Lett. A 377, 828–832 (2013)

    Article  MathSciNet  Google Scholar 

  25. Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    Book  Google Scholar 

  26. Liu, Q.P.: A note on supersymmetric two-boson equation. Commun. Theor. Phys. 25, 505–508 (1996)

    Article  MathSciNet  Google Scholar 

  27. Zhang, M.X., Liu, Q.P., Shen, Y.L., Wu, K.: Bilinear approach to \(N = 2\) supersymmetric KdV equations. Sci. China Ser. A, Math. 52, 1973–1981 (2009)

    Article  MathSciNet  Google Scholar 

  28. Laberge, C.A., Mathieu, P.: \(N = 2\) superconformal algebra and integrable \(O(2)\) fermionic extensions of the Korteweg-de Vries equation. Phys. Lett. B 215, 718–722 (1988)

    Article  MathSciNet  Google Scholar 

  29. Labelle, P., Mathieu, P.: A new \(N=2\) supersymmetric Korteweg-de Vries equation. J. Math. Phys. 32, 923–927 (1991)

    Article  MathSciNet  Google Scholar 

  30. Xue, L., Levi, D., Liu, Q.P.: Supersymmetric KdV equation: Darboux transformation and discrete systems. J. Phys. A, Math. Theor. 46, 502001 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundations of China (Grant Nos. 11501312, 12175111, 11931017, 11871471), the K. C. Wong Magna Fund in Ningbo University, the Yue Qi Outstanding Scholar Project, China University of Mining & Technology, Beijing (Grant No. 00-800015Z1177) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, P.R. China

    Xiao-Xing Niu

  2. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, P.R. China

    Xiao-Xing Niu & Q. P. Liu

  3. Department of Applied Mathematics, Ningbo University, Ningbo, 315211, P.R. China

    Lingling Xue

Authors
  1. Xiao-Xing Niu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Q. P. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lingling Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lingling Xue.

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

Niu, XX., Liu, Q.P. & Xue, L. Darboux Transformations for the Supersymmetric Two-Boson Hierarchy. Acta Appl Math 180, 12 (2022). https://doi.org/10.1007/s10440-022-00511-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-022-00511-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation