New Lax integrable hierarchy and its Liouville integrable bi-Hamiltonian structure associated an isospectral problem with an arbitrary function

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.