We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation of Ablowitz-Ladik by means of the inverse scattering transform and the Deift-Zhou nonlinear steepest descent method. The leading term is a sum of two terms that oscillate with decay of order t −1/2 .

Abstract. 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.

We consider nonlinear singular partial differential equations of the form (tDt~~p(x))u = t

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We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schrödinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.× sinh(2α 1 )sech[2α 1 (n + 1) − 2v 1 t − θ 1 ].