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High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand-Levitan-Marchenko equation

Sergey Medvedev, Irina Vaseva, Mikhail Fedoruk

TL;DR

The paper tackles the accurate solution of the Gelfand-Levitan-Marchenko equation underlying the inverse nonlinear Fourier transform for the nonlinear Schrödinger equation. It introduces the High-Order Generalized Toeplitz Inner-Bordering (HGTIB) algorithm that leverages a block Toeplitz inner-bording framework, Gregory quadrature, and the Woodbury formula to achieve up to 6th–7th order accuracy while exploiting near Toeplitz structure for speed. Numerical experiments show that one-edge Gregory weights maintain accuracy with substantial speedups, with G6d and G6 delivering the best accuracy and efficiency compared to the 2nd order TIB on coarse grids. The approach enables precise reconstruction of the potential for continuous spectra and supports hybrid methods with the Darboux transform, with potential extensions to the vector NLSE such as the Manakov system.

Abstract

We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.

High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand-Levitan-Marchenko equation

TL;DR

The paper tackles the accurate solution of the Gelfand-Levitan-Marchenko equation underlying the inverse nonlinear Fourier transform for the nonlinear Schrödinger equation. It introduces the High-Order Generalized Toeplitz Inner-Bordering (HGTIB) algorithm that leverages a block Toeplitz inner-bording framework, Gregory quadrature, and the Woodbury formula to achieve up to 6th–7th order accuracy while exploiting near Toeplitz structure for speed. Numerical experiments show that one-edge Gregory weights maintain accuracy with substantial speedups, with G6d and G6 delivering the best accuracy and efficiency compared to the 2nd order TIB on coarse grids. The approach enables precise reconstruction of the potential for continuous spectra and supports hybrid methods with the Darboux transform, with potential extensions to the vector NLSE such as the Manakov system.

Abstract

We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.
Paper Structure (4 sections, 30 equations, 4 figures)

This paper contains 4 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Gelfand-Levitan-Marchenko equation
  3. Numerical experiments
  4. Conclusion

Figures (4)

  • Figure 1: Left reflection coefficient $l(\xi)$ for the chirped hyperbolic secant ($A = 5.2$, $C = 4$) in the case of anomalous (left) and normal (right) dispersion.
  • Figure 2: Comparison of schemes with weight coefficients applying for both edges of the interval of integration and schemes with weight coefficients applying only for one edge. Root mean squared error (\ref{['err']}) with respect to the number of subintervals $M$ (left) and to the execution time trade-off (right) in the case of anomalous (top row) and normal (bottom row) dispersion.
  • Figure 3: Approximation order in the case of anomalous (left) and normal (right) dispersion.
  • Figure 4: The error $\epsilon(t)$ (\ref{['err']}) for different number of subintervals $M$ and different schemes in the case of anomalous (top row) and normal (bottom row) dispersion.