TY - JOUR

T1 - Direct scattering transform for Korteweg-de Vries solitons: Robust high-order numerical methods

AU - Gudko, A. S.

AU - Gelash, A. A.

AU - Mullyadzhanov, R. I.

PY - 2026

Y1 - 2026

N2 - We develop a family of high-order numerical schemes for solving the direct scattering problem for the Lax operator of the integrable one-dimensional Korteweg-de Vries equation, using the Magnus expansion and the inverse scattering transform theory of solitons. The algorithms demonstrate convergence for the second‑, fourth‑, and sixth‑order schemes for wave fields with up to 128 solitons provided by the dressing method, highlighting the need for high‑order schemes to correctly identify soliton eigenvalues and norming constants in particular. We demonstrate that, for a fixed target accuracy in computing the norming constants of solitons, high-order schemes can provide an order-of-magnitude speedup in computations. We show that scattering coefficients must be computed using high‑precision arithmetic to recover soliton norming constants, despite anomalous errors that act as a theoretically proven instability within the direct scattering transform framework for large computational domains. The approach exhibits robust performance for large random fields, providing the complete set of scattering data, including soliton amplitude and position parameters. This opens new avenues for the analysis of experimental and numerical data, with a particular focus on statistical problems and the identification of coherent structures in turbulent wave fields.

AB - We develop a family of high-order numerical schemes for solving the direct scattering problem for the Lax operator of the integrable one-dimensional Korteweg-de Vries equation, using the Magnus expansion and the inverse scattering transform theory of solitons. The algorithms demonstrate convergence for the second‑, fourth‑, and sixth‑order schemes for wave fields with up to 128 solitons provided by the dressing method, highlighting the need for high‑order schemes to correctly identify soliton eigenvalues and norming constants in particular. We demonstrate that, for a fixed target accuracy in computing the norming constants of solitons, high-order schemes can provide an order-of-magnitude speedup in computations. We show that scattering coefficients must be computed using high‑precision arithmetic to recover soliton norming constants, despite anomalous errors that act as a theoretically proven instability within the direct scattering transform framework for large computational domains. The approach exhibits robust performance for large random fields, providing the complete set of scattering data, including soliton amplitude and position parameters. This opens new avenues for the analysis of experimental and numerical data, with a particular focus on statistical problems and the identification of coherent structures in turbulent wave fields.

KW - Inverse scattering transform

KW - Korteweg-de vries equation

KW - Lax operator

KW - Nonlinear fourier transform

KW - Solitons

UR - https://www.mendeley.com/catalogue/ec87c985-a833-38a9-8d84-230ecdc157fc/

UR - https://www.scopus.com/pages/publications/105052709174

U2 - 10.1016/j.cnsns.2026.110890

DO - 10.1016/j.cnsns.2026.110890

M3 - Article

VL - 163

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

ER -