Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schr\"odinger Equation II

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schr"odinger equation. If $\mathrm{\mid }n\mathrm{\mid }<2t$, we have decaying oscillation of order $O(t^{-1\mathrm{/}2})$ as was proved in our previous paper. Near $\mathrm{\mid }n\mathrm{\mid }=2t$, the behavior is decaying oscillation of order $O(t^{-1\mathrm{/}3})$ and the coefficient of the leading term is expressed by the Painlev'e II function. In $\mathrm{\mid }n\mathrm{\mid }>2t$, the solution decays more rapidly than any negative power of $n$.