Virasoro symmetries of Multi-Component Gelfand-Dickey systems

TL;DR

This paper explores the symmetries and tau functions of multi-component Gelfand-Dickey hierarchies, revealing differences in symmetry generators between B and C types and their implications for integrable systems.

Contribution

It introduces the analysis of additional symmetries in multi-component Gelfand-Dickey hierarchies, including B and C types, and clarifies the differences in their symmetry generators.

Findings

The symmetry generator differs between B and C types.

Surviving additional flows have the same form in both types.

The hierarchy includes classical integrable systems like multi-component KdV and Boussinesq.

Abstract

In this paper, we mainly study the additional symmetry and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.