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Abstract and Applied Analysis
Volume 2014, Article ID 678725, 7 pages
Research Article

Nonlinear Integrable Couplings of Levi Hierarchy and WKI Hierarchy

1Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China
2School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China
3Graduate School, University of Chinese Academy of Sciences, Beijing 100049, China
4Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China
5College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received 11 June 2014; Accepted 2 July 2014; Published 14 July 2014

Academic Editor: Yufeng Zhang

Copyright © 2014 Zhengduo Shan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


With the help of the known Lie algebra, a type of new 8-dimensional matrix Lie algebra is constructed in the paper. By using the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the Wadati-Konno-Ichikawa (WKI) hierarchy are worked out, which are different from the linear integrable couplings. Based on the variational identity, the Hamiltonian structures of the above hierarchies are derived.