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.