Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing Ablowitz-Ladik system, revealing soliton dominance, oscillatory effects, and Painlevé-type asymptotics in different space-time regions using Riemann-Hilbert and steepest descent methods.

Contribution

It introduces a comprehensive asymptotic analysis of the Ablowitz-Ladik system, including soliton resolution and Painlevé asymptotics, via advanced Riemann-Hilbert problem techniques.

Findings

Solitons dominate in certain sectors for large time.

Oscillatory behavior influences asymptotics near specific regions.

Painlevé transcendents describe transition zone asymptotics.

Abstract

We investigate the soliton resolution and Painlev\'e asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value problem, which is further reformulated as a Riemann-Hilbert problem with higher-order poles. Using Fredholm theory, the Riemann-Hilbert problem with the jump contour consisting of three circles centered around the origin is uniquely solved. Then, by performing a $\bar\partial$-nonlinear steepest descent method to the Riemann-Hilbert problem, we obtain the asymptotic approximation to the solution of the focusing Ablowitz-Ladik system for large time in different space-time regions of the $(n,t)$-half plane. In the sectors $\{(n,t): n /(2t) <-M_0 \}$ and $\{(n,t): n /(2t) >M_0 \}$, where $M_0$ is a positive constant, the leading order asymptotics is dominated by the solitons; while in the sector $\{(n,t): |n /(2t) -1 <M_0^{-1} \}$, the long-time asymptotics is influenced by both the solitons and the oscillations; In the two transition zones $\{(n,t): |n /(2t)+1|t^{2/3} <C \}$ and $\{(n,t): |n /(2t)-1|t^{2/3} <C \}$ with $C$ being a positive constant, we find the Painlev\'e-type asymptotics which can be expressed in terms of the solution of the second Painlev\'e transcendents.