Skip to main content
SpringerLink
Log in

Upon Generating (2+1)-dimensional Dynamical Systems

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Under the framework of the Adler-Gel’fand-Dikii(AGD) scheme, we first propose two Hamiltonian operator pairs over a noncommutative ring so that we construct a new dynamical system in 2+1 dimensions, then we get a generalized special Novikov-Veselov (NV) equation via the Manakov triple. Then with the aid of a special symmetric Lie algebra of a reductive homogeneous group G, we adopt the Tu-Andrushkiw-Huang (TAH) scheme to generate a new integrable (2+1)-dimensional dynamical system and its Hamiltonian structure, which can reduce to the well-known (2+1)-dimensional Davey-Stewartson (DS) hierarchy. Finally, we extend the binormial residue representation (briefly BRR) scheme to the super higher dimensional integrable hierarchies with the help of a super subalgebra of the super Lie algebra sl(2/1), which is also a kind of symmetric Lie algebra of the reductive homogeneous group G. As applications, we obtain a super 2+1 dimensional MKdV hierarchy which can be reduced to a super 2+1 dimensional generalized AKNS equation. Finally, we compare the advantages and the shortcomings for the three schemes to generate integrable dynamical systems.

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.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equation. Invent. Math. 50, 219 (1979)

    Article  MATH  ADS  Google Scholar 

  2. Gel’fand, I.M., Dikii, L.A.: Collected Works. Springer, New York (1990)

    Google Scholar 

  3. Dorfamn, I.Ya., Fokas, A.S.: Hamiltonian theory over noncommutative rings and integrability in multidimensions. J. Math. Phys. 33, 2504 (1992)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Athorne, C., Dorfman, I.Ya.: The Hamiltonian structure of the (2+1)-dimensional Ablowitz-Kaup-Newell-Segur hierarchy. J. Math. Phys. 34, 3507 (1993)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. Tu, G.Z., Andrushkiw, R.I., Huang, X.C.: A trace identity and its application to integrable systems of 1+2 dimensions. J. Math. Phys. 32, 1900 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Zhang, Y.F., Rui, W.J.: On generating (2+1)-dimensional hierarchies of evolution equations. Commun. Nonlinear Sci Numer. Simulat. 19, 3454 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  7. Zhang, Y.F., Rui, W.J., Tam, H.W.: An (2+1)-dimensional expanding model of the Davey-Stewartson hierarchy as well as its Hamiltonian structure. Discontinuity, Nonlinearity, Complex. 3, 427 (2014)

    Article  MATH  Google Scholar 

  8. Zhang, Y.F., Gao, J., Wang, G.M.: Two (2+1)-dimensional hierarchies of evolution equations and their hamiltonian structures. Appl. Math. Comput. 243, 601 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Zhang, Y.F., Zhao, Z.L., Wang, G.M.: On generating linear and nonlinear integrable systems with variable coefficients. Appl. Math. Comput. 244, 672 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Tu, G.Z., Feng, B.L., Zhang, Y.F.: The residue and binormial representation of (2+1)-dimensional AKNS hierarchy. J. Weifang Univ. 14, 1 (2014)

    Google Scholar 

  11. Zhang, Y.F., Wu, L.X., Rui, W.J.: A corresponding Lie algebra of a reductive homogeneous group and its applications. Commun. Theor. Phys. 63, 535–548 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. Athorne, C., Fordy, A.: Integrable equations in (2+1)-dimensions associated with symmetric and homogeneous spaces. J. Math. Phys. 28, 2018 (1987)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Zhang, Y.F., Zhang, H.Q.: A direct method for integrable couplings of TD hierarchy. J. Math. Phys. 43, 466 (2002)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Tu, G.Z.: The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J. Math. Phys. 30, 330 (1989)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Ma, W.X.: A hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chin. J. Contemp. Math. 13, 79 (1992)

    MathSciNet  Google Scholar 

  16. Hu, X.B.: An approach to generate super-extensions of integrable systems. J. Phys. A 30, 619 (1997)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Cao, C.W., Wu, Y.T., Geng, X.G.: On quasi-periodic solutions of the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation. Phys. Lett. A 256, 59 (1999)

    Article  ADS  Google Scholar 

  18. Zhang, Y.F., Rui, W.J.: A few super-integrable hierarchies and some reductions, super-Hamiltonian structures. Rep. Math. Phys. 75, 231 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  19. Zhou, R.G.: A Darboux transformation of the sl(2/1) super KdV hierarchy and a super lattice potential KdV equation. Phys. Lett. A 378, 1816 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  20. Geng, X.G., Wu, L.H.: A new super-extension of the KdV hierarchy. Appl. Math. Lett. 23, 716 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wadati, M.: Invariances and conservation laws of the Korteweg-de Vries equation. Stud. Appl. Math. 59, 153 (1978)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. Wadati, M., Toda, M.: The exact solution of the Korteweg-de Vries equation. J. Phys. Soc. Jpn. 32, 1403 (1972)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work was supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014) and the National Natural Science Foundation of China (grant No. 11371361) as well as the Natural Science Foundation of Shandong Province (grant No. ZR2013AL016).

Yufeng Zhang is grateful to professor Tu Guizhang for his guidance and help!

Author information

Authors and Affiliations

  1. College of Sciences, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Yufeng Zhang

  2. School of Environment Science and Spatial Informatics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Yang Bai & Lixin Wu

Authors
  1. Yufeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lixin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yufeng Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Bai, Y. & Wu, L. Upon Generating (2+1)-dimensional Dynamical Systems. Int J Theor Phys 55, 2837–2856 (2016). https://doi.org/10.1007/s10773-016-2916-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-016-2916-z

Keywords

Search

Navigation