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A generalized AKNS hierarchy and its bi-Hamiltonian structures. (English) Zbl 1071.37048

Summary: First, we construct a new isospectral problem with 8 potentials. And then, a new Lax pair is presented. By making use of Tu’s scheme, a class of new soliton hierarchies of equations is derived, which is integrable in the sense of Liouville and possesses bi-Hamiltonian structures. After making some reductions, the well-known AKNS hierarchy and other hierarchies of evolution equations are obtained. Finally, in order to illustrate that the soliton hierarchy obtained in the paper possesses bi-Hamiltonian structures exactly, we prove that the linear combination of two Hamiltonian operators admitted is also a Hamiltonian operator constantly. We point out that two Hamiltonian operators obtained of the system are directly derived from a recurrence relation, not from a recurrence operator.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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