Soliton solutions by Darboux transformation and some reductions for a new Hamiltonian lattice hierarchy

Abstract

In this paper, we start from the discrete spectral problem and construct a lattice hierarchy by properly choosing an auxiliary spectral problem (V) over tilde (n)(m), 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 epsilon((1))(n) = 0 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.