Integrable Discretizations of Chiral Models

TL;DR

This paper develops a method to create integrable discrete versions of chiral models, including Toda lattices, by deforming differential calculus, and constructs associated Lax pairs and Bäcklund transformations.

Contribution

It introduces a novel approach to discretize chiral models while preserving integrability, extending conservation laws and solution techniques to discrete settings.

Findings

Discrete chiral models with conservation laws

Discrete Toda lattice derived from continuum wave equation

Construction of Lax pairs and Bäcklund transformations for discrete models

Abstract

A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and B\"acklund transformations for the class of models considered in this work.