On Soliton Equations of Exceptional Type

Abstract

The main purpose of this paper is to present an explicit formula for the general hierarchy of soliton equations constructed by Kac-Wakimoto from the basic representation of an arbitrary affine Kac-Moody algebra. The results turn out that the differential operators of the corresponding Hirota bilinear equations can be written explicitly in terms of skew Schur functions for both principal and homogeneous hierarchies. The principal hierarchy includes the classical KP and KdV equations. The homogeneous hierarchy turns out to be related to the classical non-linear Schrodinger equation for type A(1)1 and to the classical 2-dimensional Toda lattice equation for type A(1)2.