Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy |
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Authors: | Xianguo Geng H.H. Dai |
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Affiliation: | a Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People's Republic of China b Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People's Republic of China |
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Abstract: | The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map. |
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Keywords: | Discrete Ablowitz-Ladik hierarchy Nonlinearization of the Lax pairs |
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