New integrable differential-difference systems: Lax pairs, bilinear forms and soliton solutions

Abstract

Two new integrable differential-difference systems with their Lax pairs are proposed. By the dependent variable transformations, these integrable lattices can be transformed into bilinear equations. With the assistance of Mathematica, three-soliton solutions are explicitly obtained. We have also shown that these lattices can be obtained from a special case of the coupled bilinear equations under reduction. Furthermore a bilinear Bäcklund transformation and the corresponding nonlinear superposition formula concerning the coupled bilinear equations are presented. Besides, it is also illustrated that the y-flow of these coupled bilinear equations can be transformed into a lattice previously derived by the authors. Starting from the corresponding bilinear Bäcklund transformation, its corresponding Lax pair is obtained.