Long-time asymptotic analysis for defocusing Ablowitz-Ladik system with initial value in lower regularity

TL;DR

This paper analyzes the long-time behavior of the defocusing Ablowitz-Ladik system with initial data in lower regularity spaces, revealing asymptotic formulas and decay properties using Riemann-Hilbert problem techniques.

Contribution

It extends previous spectral results to establish long-time asymptotics for lower regularity initial data in the Ablowitz-Ladik system.

Findings

Solution exhibits Zakharov-Manakov type asymptotics when |n/2t| ≤ 1.

Solution decays rapidly to zero when |n/2t| ≥ 1.

Results apply to initial data in lower regularity Sobolev spaces.

Abstract

Recently, we have given the $l^2$ bijectivity for defocusing Ablowitz-Ladik systems in the discrete Sobolev space $l^{2,1}$ by inverse spectral method. Based on these results, the goal of this article is to investigate the long-time asymptotic property for the initial-valued problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity. The main idea is to perform proper deformations and analysis to the corespondent Riemann-Hilbert problem with the unit circle as the jump contour $\Sigma$. As a result, we show that when $|\frac{n}{2t}|\le 1<1$, the solution admits Zakharov-Manakov type formula, and when $|\frac{n}{2t}|\ge 1>1$, the solution decays fast to zero.