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 $|n|<2t$, 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 Painlev\'e II function. In $|n|>2t$, the solution decays more rapidly than any negative power of $n$. 
Hideshi Yamane
math-ph math.MP