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

TL;DR

This paper demonstrates how Virasoro and W-constraints naturally emerge in the generalized KdV hierarchy using the Lax operator approach, linking symmetries to the string equation.

Contribution

It provides a direct Lax operator method to derive Virasoro and W-constraints in the p-reduced KP hierarchy, including KdV and Boussinesq cases.

Findings

Virasoro and W-constraints follow from the string equation

Method applies to KdV and Boussinesq hierarchies

Potential generalization to higher KdV hierarchies

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.