Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem

TL;DR

This paper introduces two fourth-order exponential finite-difference algorithms for solving the Zakharov-Shabat system, conserving invariants and enabling fast computation, advancing numerical methods for integrable systems.

Contribution

The paper presents novel fourth-order exponential schemes that conserve invariants and incorporate spectral parameters efficiently for the Zakharov-Shabat problem.

Findings

Both schemes achieve fourth-order accuracy.

The second scheme allows fast computation due to spectral parameter placement.

Invariant conservation is maintained by both algorithms.

Abstract

We propose two finite-difference algorithms of fourth order of accuracy for solving the initial problem of the Zakharov-Shabat system. Both schemes have the exponential form and conserve quadratic invariant of Zakharov-Shabat system. The second scheme contains the spectral parameter in exponent only and allows to apply the fast computational algorithm.