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Journal of Applied Nonlinear Dynamics 1(1) (2011) 1--28 | DOI:10.5890/JAND.2011.12.001

Yufeng Zhang$^{1}$; Wen-XiuMa$^{2}$

$^{1}$ College of Science, China University of Mining and Technology, Xuzhou 221116, P. R. China

$^{2}$ Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA

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Abstract

Based on two higher-dimensional extensions of Lie algebras, three kinds of specific Lie algebras are introduced. Upon constructing proper loop algebras, six isospectral matrix spectral problems are presented and they yield nonlinear integrable couplings of the Ablowitz- Kaup-Newell-Segur hierarchy, the Broer-Kaup hierarchy and the Kaup-Newell hierarchy. Especially, the reduced cases of the resulting integrable couplings give nonlinear integrable couplings of the nonlinear Schrödinger equation and the classical Boussinesq equation. Two linear functionals are introduced on two loop algebras of dimension 6 and Hamiltonian structures of the obtained nonlinear integrable couplings are worked out by employing the associated variational identity. The proposed approach can also be used to generate nonlinear integrable couplings for other integrable hierarchies.

Acknowledgments

The work was supported in part by the State Administration of Foreign Experts Affairs of China, the National Natural Science Foundation of China (Nos.6172147 and 11071159), Chunhui Plan of the Ministry of Education of China, the Natural Science Foundation of Shanghai, Shanghai Leading Academic Discipline Project (No. J50101) and the fundamental Research Funds of the Central University (2010LKSX08) and the Natural Science Foundation of Liaoning Province(20092171).

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