### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 020, 17 pages      arXiv:1407.5751      https://doi.org/10.3842/SIGMA.2015.020

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

Hideshi Yamane
Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan

Received September 06, 2014, in final form March 03, 2015; Published online March 08, 2015

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 $n$.

Key words: discrete nonlinear Schrödinger equation; nonlinear steepest descent; Painlevé equation.

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