Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons

We investigate the soliton resolution and Painlev\'e asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value problem, which is further reformulated as a Riemann-Hilbert problem with higher-order poles. Using Fredholm theory, the Riemann-Hilbert problem with the jump contour consisting of three circles centered around the origin is uniquely solved. Then, by performing a $\bar\partial$-nonlinear steepest descent method to the Riemann-Hilbert problem, we obtain the asymptotic approximation to the solution of the focusing Ablowitz-Ladik system for large time in different space-time regions of the $(n,t)$-half plane. In the sectors $\{(n,t): n /(2t) <-M_0 \}$ and $\{(n,t): n /(2t) >M_0 \}$, where $M_0$ is a positive constant, the leading order asymptotics is dominated by the solitons; while in the sector $\{(n,t): |n /(2t) -1 <M_0^{-1} \}$, the long-time asymptotics is influenced by both the solitons and the oscillations; In the two transition zones $\{(n,t): |n /(2t)+1|t^{2/3} <C \}$ and $\{(n,t): |n /(2t)-1|t^{2/3} <C \}$ with $C$ being a positive constant, we find the Painlev\'e-type asymptotics which can be expressed in terms of the solution of the second Painlev\'e transcendents.