Global classification of two-component approximately integrable evolution equations

TL;DR

This paper provides a comprehensive classification of two-component evolution equations with specific symmetry properties, utilizing advanced mathematical tools like symbolic calculus, number theory, and algorithms.

Contribution

It introduces a global classification framework for these equations, combining symbolic calculus, number theory, and computational algorithms to identify integrability features.

Findings

Classification of equations with infinite approximate symmetries

Identification of algebraic structures using symbolic calculus

Application of number theory to solve diophantine equations in this context

Abstract

We globally classify two-component evolution equations, with homogeneous diagonal linear part, admitting infinitely many approximate symmetries. Important ingredients are the symbolic calculus of Gel'fand and Dikii, the Skolem-Mahler-Lech theorem, results on diophantine equations in roots of unity by F. Beukers, and an algorithm of C.J. Smyth.