Fast Computation of the Direct Scattering Transform by Fourth Order Conservative Multi-Exponential Scheme

TL;DR

This paper introduces a fourth-order multi-exponential scheme for the Zakharov-Shabat system that enables fast computation of the direct scattering transform while preserving quadratic invariants for real spectral parameters.

Contribution

It develops a novel fourth-order scheme based on exponential operator factorization, improving computational efficiency and invariant conservation in scattering transform calculations.

Findings

Allows fast computation for many spectral parameters

Exactly conserves quadratic invariants for real spectral parameters

Uses a product of 13 exponential operators for high-order accuracy

Abstract

A fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The scheme represents a product of 13 exponential operators. The construction of the scheme is based on a fourth-order three-exponential scheme, which contains only one exponent with a spectral parameter. This exponent is factorized to the fourth-order with the Suzuki formula of 11 exponents. The obtained scheme allows the use of a fast algorithm in calculating the initial problem for a large number of spectral parameters and conserves the quadratic invariant exactly for real spectral parameters.