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

TL;DR

This paper investigates how integrable wave dynamics persist in the focusing Discrete Gross-Pitaevskii equation, demonstrating long-term soliton stability and proximity to the Ablowitz-Ladik lattice, supported by theoretical estimates and numerical simulations.

Contribution

It provides new estimates for solution distances in weighted spaces and shows the long-term persistence of small-amplitude solitons in the discrete model.

Findings

Solutions remain close over long times in weighted metrics.

Small-amplitude bright solitons are robust under weak harmonic traps.

Numerical simulations confirm theoretical proximity and soliton robustness.

Abstract

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.