On the Drinfeld-Sokolov Hierarchies of D type

Abstract

We extend the notion of pseudo-differential operators that are used to represent the Gelfand-Dickey hierarchies, and obtain a similar representation for the full Drinfeld-Sokolov hierarchies of $D_n$ type. By using such pseudo-differential operators we introduce the tau functions of these bi-Hamiltonian hierarchies, and prove that these hierarchies are equivalent to the integrable hierarchies defined by Date-Jimbo-Kashiware-Miwa and Kac-Wakimoto from the basic representation of the Kac-Moody algebra $D_n^{(1)}$.