Stability of Schur’s iterates and fast solution of the discrete integrable NLS

We prove a sharp stability estimate for Schur iterates of contractive analytic functions in the open unit disk. We then apply this result in the setting of the inverse scattering approach and obtain a fast algorithm for solving the discrete integrable nonlinear Schrödinger equation (Ablowitz–Ladik equation) on the integer lattice, \mathbb{Z} . We also give a self-contained introduction to the theory of the nonlinear Fourier transform from the perspective of Schur functions and orthogonal polynomials on the unit circle.