Virasoro symmetries of Multi-Component Gelfand-Dickey systems

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.

arXiv comment23 Pages, to appear in Theoretical and Mathematical Physics

References in corpus (9)

• On additional symmetries of the KP hierarchy and Sato's Bäcklund transformation
• Additional symmetries of KP, Grassmannian, and the string equation
• Virasoro Symmetry of Constrained KP Hierarchies
• Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-Backlund Solutions
• Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• The Lax Operator Approach for the Virasoro and the W-Constraints in the Generalized KdV Hierarchy
• KdV type hierarchies, the string equation and $W_{1+\infty }$ constraints
• Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies