Linear integral equations and two-dimensional Toda systems

TL;DR

This paper presents a unified linear integral equation framework for analyzing two-dimensional Toda systems linked to various infinite-dimensional Lie and Kac--Moody algebras, enabling solution construction and revealing integrability structures.

Contribution

It introduces a comprehensive linear integral equation approach that unifies the understanding of Toda systems associated with multiple algebraic structures and facilitates solution derivation.

Findings

Unified linear integral equations for Toda systems

Reconstruction of Lax pairs and solutions

Construction of generalized Cauchy matrix solutions

Abstract

The direct linearisation framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional Lie algebras $A_\infty$, $B_\infty$ and $C_\infty$, as well as the Kac--Moody algebras $A_{r}^{(1)}$, $A_{2r}^{(2)}$, $C_{r}^{(1)}$ and $D_{r+1}^{(2)}$ for arbitrary integers $r\in\mathbb{Z}^+$, from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearising type solution potentially leading to the universal solution space, for each class of the two-dimensional Toda system. As particular applications of this framework to the two-dimensional Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalised Cauchy matrix solutions.