Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions

TL;DR

This paper presents a comprehensive list of forty third-order integrable two-field evolutionary systems, explores their interconnections via differential substitutions, and demonstrates that all systems derive from just two core systems, including examples of zero curvature representations.

Contribution

The paper introduces a complete list of integrable systems, establishes their interrelations through differential substitutions, and shows all systems originate from two fundamental systems.

Findings

All systems can be derived from two core systems.

Differential substitutions connect various systems.

Zero curvature representations with 4x4 matrices are provided.

Abstract

A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of them. Examples of zero curvature representations with $4 \times 4$ matrices are presented.