Long-time asymptotic analysis for defocusing Ablowitz-Ladik system with initial value in lower regularity

Recently, we have given the $l^2$ bijectivity for defocusing Ablowitz-Ladik systems in the discrete Sobolev space $l^{2,1}$ by inverse spectral method. Based on these results, the goal of this article is to investigate the long-time asymptotic property for the initial-valued problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity. The main idea is to perform proper deformations and analysis to the corespondent Riemann-Hilbert problem with the unit circle as the jump contour $\mathrm{\Sigma }$. As a result, we show that when $\mathrm{\mid }\frac{n}{2t}\mathrm{\mid }\le 1<1$, the solution admits Zakharov-Manakov type formula, and when $\mathrm{\mid }\frac{n}{2t}\mathrm{\mid }\ge 1>1$, the solution decays fast to zero.