Abstract
In this paper, we start from the discrete spectral problem and construct a lattice hierarchy by properly choosing an auxiliary spectral problem , which can reduce to the Volterra hierarchy, the Ablowitz–Ladik hierarchy, positive and negative lattice hierarchies and a new hierarchy. The new hierarchy is integrable in involutory Lax's sense and possesses multi-Hamiltonian structure. In addition, the Darboux transformation of the lattice hierarchy is obtained when the freely adjustable function and m=1. Then some soliton solutions are obtained by using Darboux transformation. This method is also suitable for other more general spectral problems in mathematics and physics.
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