标题： 离散Gross-Pitaevskii方程中可积波动力学的持久性：聚焦情形 显示英文标题

标题： Persistence of integrable wave dynamics in the Discrete Gross--Pitaevskii equation: the focusing case

摘要： 在非线性薛定谔方程框架内，我们之前的研究探讨了可积系统与非可积系统动力学的接近性。本文进一步研究了这一现象，将聚焦的离散Gross-Pitaevskii方程与Ablowitz-Ladik格子进行比较。由于存在谐波陷阱，需要在加权空间中研究Ablowitz-Ladik格子。我们建立了适当度量下解之间距离的估计，并全面描述了这种距离的一般初始数据潜在演化过程。这些结果适用于广泛的非线性薛定谔模型，包括离散和偏微分方程。对于离散Gross-Pitaevskii方程，它们保证了小振幅亮孤子在长时间内的持久性，这得益于Ablowitz-Ladik格子解析解的驱动，尤其是在弱谐波陷阱存在的情况下。数值模拟证实了关于系统在长时间内动力学接近性的理论预测。它们还揭示了即使弱谐波陷阱的影响逐渐显著，孤子也表现出显著的稳健性，导致孤子轨道呈弯曲状。 显示英文摘要

摘要： Expanding upon our prior findings on the proximity of dynamics between integrable and non-integrable systems within the framework of nonlinear Schr\"odinger equations, we examine this phenomenon for the focusing Discrete Gross-Pitaevskii equation in comparison to the Ablowitz-Ladik lattice. The presence of the harmonic trap necessitates the study of the Ablowitz-Ladik lattice in weighted spaces. We establish estimates for the distance between solutions in the suitable metric, providing a comprehensive description of the potential evolution of this distance for general initial data. These results apply to a broad class of nonlinear Schr\"odinger models, including both discrete and partial differential equations. For the Discrete Gross-Pitaevskii equation, they guarantee the long-term persistence of small-amplitude bright solitons, driven by the analytical solution of the AL lattice, especially in the presence of a weak harmonic trap. Numerical simulations confirm the theoretical predictions about the proximity of dynamics between the systems over long times. They also reveal that the soliton exhibits remarkable robustness, even as the effects of the weak harmonic trap become increasingly significant, leading to the soliton's curved orbit.