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

TL;DR

This paper analyzes the long-time behavior of solutions to the defocusing integrable discrete nonlinear Schrödinger equation, revealing different decay rates and oscillation behaviors depending on the relation between spatial and temporal variables.

Contribution

It provides a detailed asymptotic description of solutions, including the emergence of Painlevé II functions near critical points, extending previous results to a broader regime.

Findings

Decaying oscillation of order O(t^{-1/2}) for |n|<2t

Oscillation of order O(t^{-1/3}) near |n|=2t with Painlevé II coefficient

Rapid decay of solutions for |n|>2t

Abstract

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 $n$.