Remarks on the Additional Symmetries and W-Constraints in the Generalized KdV Hierarchy

TL;DR

This paper investigates the additional symmetries of the p-reduced KP hierarchy, showing they form a W-algebra only under specific conditions involving the operators L and M.

Contribution

It explicitly demonstrates the conditions under which the generators of additional symmetries satisfy a closed W-algebra in the generalized KdV hierarchy.

Findings

Additional symmetries generate a W-algebra only with certain operator constraints.

The operators L and M satisfy specific residue conditions.

The symmetry algebra closes under the imposed conditions.

Abstract

Additional symmetries of the $p$-reduced KP hierarchy are generated by the Lax operator $L$ and another operator $M$, satisfying $res (M^n L^{m+n/p})$ = 0 for $1 \leq n \leq p-1$ and $m \geq -1$ with the condition that ${\partial L \over {\partial t_{kp}}}$ = 0, $k$ = 1, 2,..... We show explicitly that the generators of these additional symmetries satisfy a closed and consistent W-algebra only when we impose the extra condition that ${\partial M \over {\partial t_{kp}}} = 0$.