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

R. V. Bessonov, P. V. Gubkin

Abstract

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.

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

Abstract

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, . 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.
Paper Structure (18 sections, 31 theorems, 187 equations)

This paper contains 18 sections, 31 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Schur's algorithm
  3. AL: statement of the problem
  4. AL: localization
  5. AL: compactly supported initial data
  6. AL: algorithm for Problem \ref{['Problem']}
  7. AL: historical remarks and motivation
  8. The nonlinear Fourier transform
  9. Schur's algorithm. Proof of Theorem \ref{['t0']}.
  10. Estimates for the multipliers. Proof of Theorem \ref{['t3']}
  11. Localization. Proof of Theorem \ref{['t2']}
  12. Complexity of the algorithm
  13. The nonlinear Fourier transform. Proof of Theorem \ref{['t5']}
  14. Szegő measures and Szegő functions
  15. Reflection coefficients
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $F \in \mathcal{S}_*( \mathbb D)$, and let $\{F_n(0)\}_{n\geqslant 0}$ be its recurrence coefficients. Then where both sides are finite or infinite simultaneously.

Theorems & Definitions (34)

  • Theorem 1.1: Szegő theorem
  • Theorem 1.2
  • Theorem 1.3: Sylvester--Winebrenner theorem
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • ...and 24 more