The Lax Operator Approach for the Virasoro and the W-Constraints in the Generalized KdV Hierarchy

Abstract

We show directly in the Lax operator approach how the Virasoro and W-constraints on the $\tau$-function arise in the $p$-reduced KP hierarchy or generalized KdV hierarchy. In partiacular, we consider the KdV and Boussinesq hierarchy to show that the Virasoro and W-constraints follow from the string equation by expanding the ``additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.