Abstract
The rational integrable differential-difference equations are proposed from a different discrete spectral problem. We proved the Liouville integrability of rational integrable differential-difference equations by deriving its bi-Hamiltonian structures. Furthermore, infinitely many conservation laws of the hierarchy are listed based on the Lax pairs. We found Darboux matrices to construct their Darboux transformations (DT) of the first two nontrivial equations respectively. Explicit solutions of the first two nontrivial equations are presented and analyzed.
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