Discrete Integrable Systems And Its Integrable Coupling Systems

Discrete integrable systems and its integrable coupling systems are presented in this paper. In Section 2, firstly, two new discrete 2×2 matrix spectral problems are discussed, then we focus on two new discrete 3×3 matrix spectral problems, and by the discrete zero curvature equation, the associated Lax integrable equations are derived. By means of the trace identity the Hamiltonian structures are constructed for the equations and the Liouville integrability is also proved. In Section 3, two discrete 4×4 spectral problems are proposed and a hierarchy of discrete integrable coupling is derived by means of the method of semi-direct sums of Lie algebras. Furthermore, the Hamiltonian structures are constructed for the systems by means of the discrete variational identity, the Liouville integrabbility is also proved.