Linear integral equations and two-dimensional Toda systems

Abstract

A general framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional Lie algebras A ∞ , B ∞ and C ∞ , as well as the Kac-Moody algebras A (1) r , A (2) 2r , C (1) r and D (2) r+1 for arbitrary integers r ∈ Z + , from the aspect of a set of linear integral equations in a certain form. Our scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces a general 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.