  In this article, we investigate some problems about the initial-value problem
of the focusing Ablowitz-Ladik system, and the initial data belongs to a
discrete weighted $\ell^2$ space. On the one hand, we have proved the global
well-posedness for the initial-value problem. Utilizing the fact that the
initial data belongs to the discrete weighted $\ell^2$ space, we construct a
Riemann-Hilbert problem for the initial-value problem. By Fredholm theory, the
Riemann-Hilbert problem is uniquely solved with the jump contour consisting of
three circles that all center around the origin. On the other hand, based on
the global well-posedness and the Riemann-Hilbert problem, we allow the
solution to have the higher-order solitons, and then analyze the long-time
asymptotics for the solution in different sectors on the upper-half plane.
