Affine Lie algebras, Lax pairs and integrable discrete and continuous systems

TL;DR

This paper introduces a unified framework linking affine Lie algebras to integrable systems, presenting new discrete and continuous models, their Lax pairs, and Bäcklund transformations, expanding the understanding of integrability in mathematical physics.

Contribution

It constructs a set of integrable systems associated with affine Lie algebras, including new discrete, semi-discrete, and continuous models with their Lax pairs and Bäcklund transforms.

Findings

Established a set of six integrable systems from affine Lie algebras.

Derived Lax pairs for these systems using Zakharov-Shabat equations.

Connected discrete and continuous systems through Bäcklund transformations.

Abstract

A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of generalized Toda lattice while semi-discrete systems in the set define the Backlund transform for this Toda lattice and the fully discrete representative of the set can be obtained as a superposition of such kind Backlund transforms. Four linear Zakharov-Shabat type systems taken pairwise realize Lax pairs for these six dynamical systems.