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

Abstract

We investigate the soliton resolution and Painlevé 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 Painlevé-type asymptotics which can be expressed in terms of the solution of the second Painlevé transcendents.