Linear integral equations and two-dimensional Toda systems

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.

26 pages