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

TL;DR

This work analyzes the focusing Ablowitz-Ladik system with initial data in a discrete weighted ℓ^2 space, establishing global well-posedness in ℓ^{2,1} and reformulating the IVP as a Riemann-Hilbert problem with higher-order poles. A $ar d$-steepest descent analysis of the pure RH problem, obtained after pole removal via triangular factors, yields precise long-time asymptotics across all space-time sectors: in sectors I and III the solution is soliton-dominated with a decay of order $O(t^{-1})$, in sector II a soliton-oscillatory mix occurs with $O(t^{-3/4})$ error, and in the first and second transition zones Painlevé II type asymptotics appear, governed by a Painlevé II transcendent. The results rely on a Fredholm framework for solvability of the RH problem, a three-circle contour for the pure RH formulation, and careful local modeling via parabolic-cylinder and Painlevé II RH problems, providing a rigorous soliton-resolving picture for this discrete integrable system. Overall, the paper extends the toolkit of discrete inverse scattering to address higher-order spectral data and reveals sharp, region-wise asymptotics that mirror the continuous NLS setting while highlighting discrete-specific features and transition phenomena.

Abstract

We investigate the soliton resolution and Painlevé 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 Painlevé-type asymptotics which can be expressed in terms of the solution of the second Painlevé transcendents.

Figures (10)

• Figure 1: The different asymptotic sectors of the $(n,t)$-half plane, where $\xi =\frac{n}{2t}$.
• Figure 2: The jump contour $\tilde{\Sigma}$ consists of three circles centering at the origin, which is for RH problem \ref{['r2.7']}
• Figure 3: The signature table for $\mathrm{Re}\phi$, where $\Sigma$ is the jump contour, $\mathrm{Re}\phi>0$ on the grayed region, and $\mathrm{Re}\phi<0$ otherwise.
• Figure 4: The jump contour $\Sigma^{(2)}=\Sigma^{(2)}_1\cup\Sigma^{(2)}_2\cup\Sigma^{(2)}_3\cup\Sigma^{(2)}_4$ and regions $\Omega_1,\dots,\Omega_4$, where $\Sigma^{(2)}_i=\partial\Omega_i\cap D_+$ for $i=1,2$ and $\Sigma^{(2)}_i=\partial\Omega_i\cap D_-$ for $i=3,4$.
• Figure 5: Jump contours $\Sigma^{(2,j)}$, $j=1,2$.

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 3.1
• proof : Proof of Proposition \ref{['p2.2']}
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Lemma 4.4: Vanishing Lemma