Decomposition of the discrete Ablowitz-Ladik hierarchy

The nonlinearization approach of Lax pairs is extended to the discrete Ablowitz-Ladik hierarchy. A new symplectic map and a class of new finite-dimensional Hamiltonian systems are derived, which are further proved to be completely integrable in the Liouville sense. An algorithm to solve the discrete Ablowitz-Ladik hierarchy is proposed. Based on the theory of algebraic curves, the straightening out of various flows is exactly given through the Abel-Jacobi coordinates. As an application, explicit quasi-periodic solutions for the discrete Ablowitz-Ladik hierarchy are obtained resorting to the Riemann theta functions.