International Journal of Theoretical Physics

, Volume 55, Issue 6, pp 2837–2856 | Cite as

Upon Generating (2+1)-dimensional Dynamical Systems

  • Yufeng Zhang
  • Yang Bai
  • Lixin Wu


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.


Hamiltonian pair TAH scheme Super Lie algebra 



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!


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of SciencesChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.School of Environment Science and Spatial InformaticsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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