Two integrable lattice hierarchies and their respective Darboux transformations

Two integrable lattice hierarchies associated with two discrete matrix spectral problems are derived, and the infinitely many conservation laws of the first integrable model is obtained. Moreover, the Darboux transformations based on different Darboux matrixes (2.3.4) and (3.2.4) for the above-mentioned integrable hierarchies are established with the help of two different gauge transformations of lax pairs in this paper. As an application, the explicit solutions of the two integrable hierarchies are presented. Nonlinear integrable lattice equations [1–9] have explicit solutions with perfect mathematical and physical properties. Therefore, to obtain the exact solutions for the integrable lattice equations is always one of the most fundamental and difficult topics. There are several new approaches of the solution methods which include a transformed rational function method for constructing traveling wave solutions [10] and a multiple exp-function method for presenting multiple wave solutions [11,12]. Recently, the linear superposition principle was used to construct linear-subspaces of solutions to Hirota bilinear equations [13]. The Darboux transformation (DT) [14–25] is one of the powerful methods in finding explicit solutions of soliton equation from a trivial seed. Essentially, the DT is a special gauge transformation of solutions of Lax pairs. The key for constructing DT is to expose a kind of covariant properties that corresponding spectral problems possess. However, it is difficult to solve the lax pairs of Lattice soliton equations if the potential functions are not trivial. So, only while the isospectral matrix is suitable given, the DT is an effect way to solve the lattice equations. In the past decades, there has been significant progress in the development of DT. A kind of Darboux transformation for a Lax integrable system in 2n-dimensions was established in [26]. The binary DT was discussed in [27]. The connection between the DT and some modern operator based approaches to quantum mechanics is also outlined in [28]. DTs are set up for the (2+1)-dimensional integrable lattice equation [29] and (1+1)-dimensional systems. The DT of integra-ble lattice equations associated with 3 Â 3 matrix spectral problems are derived in [30,31]. In this paper, we would like to give the exact solutions of two integrable lattice equations by the DT method based on two different Darboux matrixes. This paper is organized as follows. In Section 2, we derive the generalized Lotka–Volterra lattice equation [32,33] and its DT. In Section 3, we deduce another integrable lattice equation [34] and its DT. In Section 4, …