p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p)

TL;DR

This paper links tau-functions of p-reduced multicomponent KP hierarchies to solutions of associated integrable PDEs, providing explicit constructions of classical W-algebras and their evolution formulas.

Contribution

It demonstrates that tau-functions generate solutions to the hierarchy and offers an explicit algorithm for constructing classical W-algebra generators.

Findings

Tau-functions produce solutions to the integrable hierarchy.

Explicit formulas for W-algebra generators and their evolution.

Algorithm for constructing classical W-algebras.

Abstract

For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W-algebra W(gl_N,p), and write down explicit formulas for evolution of these generators along the Hamiltonian flows.