Abstract
Two central and significative but difficult subjects in soliton theory and nonlinear integrable dynamic systems are to seek new Lax integrable hierarchies and their Hamiltonian structure (bi-Hamiltonian structure). In this paper, an new isospectral problem with an arbitrary function and the associated Lax integrable hierarchy of evolution equations are presented by using zero curvature equation. As a result, a representative system of the generalized derivative nonlinear Schrödinger equations with an arbitrary function in the hierarchy is obtained. Bi-Hamiltonian structure is established for the whole hierarchy based upon a pair of Hamiltonian operators and it is shown that the hierarchy is Liouville integrable. In addition, infinitely many commuting symmetries of the hierarchy are given.
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