Continuum limit for the Ablowitz--Ladik system

TL;DR

This paper proves that solutions of the Ablowitz--Ladik system converge to the cubic nonlinear Schr"odinger equation for $L^2$ initial data, including cases with multiple excited Fourier modes, establishing a continuum limit.

Contribution

It establishes the continuum limit of the Ablowitz--Ladik system to the nonlinear Schr"odinger equation for broad initial data, including multiple Fourier modes.

Findings

Solutions converge to NLS for $L^2$ data

Convergence holds for initial data exciting multiple Fourier modes

Results extend understanding of discrete-to-continuum limits

Abstract

We show that solutions to the Ablowitz--Ladik system converge to solutions of the cubic nonlinear Schr\"odinger equation for merely $L^2$ initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schr\"odinger equations.