A special integrable differential-difference equation and its related systems: Bilinear forms soliton solutions and Lax pairs

In this paper, a special integrable differential-difference equation and its related systems are studied. First of all, by using dependent variable transformations, this special lattice is transformed into two bilinear forms. As a result, the corresponding soliton solutions are obtained. A coupled set of bilinear equations is proposed and related to the same special lattice in a certain way. We also derive the C-flow and z-flow of the coupled bilinear equations. Lax pairs for the t-flow and the z-flow are given. Furthermore, a bilinear Bäcklund transformation and the corresponding nonlinear superposition formula for the coupled bilinear equations are presented. Soliton solutions to the coupled bilinear equations are derived.