Solving discrete nonlinear Shrödinger equation on $\mathbb Z$ using Schur's algorithm for analytic functions

In 1987, Y. Tsutsumi proved that the nonlinear Shrödinger equation
$$ iu'_t = -u''_{xx} \pm 2|u|^2 u, \quad x, t \in \mathbb R, \quad u\bigr\rvert_{t=0} = u_0 $$
admits a unique global weak solution for any initial datum $u_0 \in L^2(\mathbb R)$. He used a direct method based on a Strichartz estimate. This result is not available by means of the classical inverse scattering theory (IST) due to the following fundamental obstacle: the scattering data for the Dirac equation with $L^2(\mathbb R)$–potential (the auxiliary problem for NLSE) do not determine the potential uniquely. This fact was discovered in 2002 by A. Volberg and P. Yuditskii on the level of Jacobi matrices.

We discuss how to modify the IST approach to solve the discrete integrable NLS equation (Ablowitz-Ladik equation) with $\ell^2(\mathbb Z)$ initial data by means of inverse scattering. The argument is based on a new estimate for the classical Schur's algorithm for contractive analytic functions in Szegő class. It also gives a new exponentially fast numerical scheme for solving discrete NLSE. The continuous case remains open.