Three discrete integrable coupling schemes associated with relativistic Toda lattice equation

In this paper, three discrete integrable coupling schemes from three different semi-direct sums of Lie algebras are illustrated. By the discrete variational identity, three different bi-Hamiltonian structures are proposed about discrete relativistic Toda lattice hierarchies, which are derived from three different types of discrete coupling spectral problems and the three hierarchies are all reduced to the discrete relativistic Toda lattice equations. In this way, many other multiform types and classifications of other integrable couplings can also be deduced.