Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If $|n|<2t$, we have decaying oscillation of order $O(t^{-1/2})$ as was proved in our previous paper. Near $|n|=2t$, the behavior is decaying oscillation of order $O(t^{-1/3})$ and the coefficient of the leading term is expressed by the Painlevé II function. In $|n|>2t$, the solution decays more rapidly than any negative power of $n$.