Continuum limit for the Ablowitz--Ladik system

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.