Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system in optical applications

TL;DR

This paper introduces a sixth-order accurate, conservative one-step scheme for the Zakharov-Shabat system using the generalized Cayley transform, enabling fast solutions for multiple spectral parameters in optical applications.

Contribution

It develops a new family of high-order schemes based on rational approximation and the Cayley transform, improving computational efficiency for the Zakharov-Shabat system.

Findings

Achieves sixth-order accuracy in numerical solutions.

Enables fast computation for multiple spectral parameters.

Includes a special case as an exponential integrator.

Abstract

Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.