On the Drinfeld-Sokolov Hierarchies of D type

TL;DR

This paper extends pseudo-differential operator techniques to represent Drinfeld-Sokolov hierarchies of D type, introduces tau functions, and proves their equivalence to known integrable hierarchies from affine Kac-Moody algebra representations.

Contribution

It develops a pseudo-differential operator framework for D-type hierarchies and establishes their equivalence to existing integrable systems.

Findings

Representation of D-type hierarchies via pseudo-differential operators

Introduction of tau functions for these hierarchies

Proof of equivalence to hierarchies from Kac-Moody algebra representations

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 $D_n^{(1)}$.