Integrable equations and recursion operators related to the affine Lie algebras $A^{(1)}_{r}$

TL;DR

This paper derives a family of integrable equations associated with affine Lie algebras $A^{(1)}_{r}$ using Coxeter reductions, contributing to the understanding of soliton hierarchies and their algebraic structures.

Contribution

It introduces a new family of integrable equations linked to affine Lie algebras via Coxeter reductions, expanding the class of known soliton equations.

Findings

Derived equations related to $A^{(1)}_{r}$ affine Lie algebras

Presented examples and additional reductions of the equations

Connected the equations to the hierarchy of soliton equations

Abstract

We have derived a family of equations related to the untwisted affine Lie algebras $A^{(1)}_{r}$ using a Coxeter $\mathbb{Z}_{r+1}$ reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. We also give some particular examples and impose additional reductions.