Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type

TL;DR

This paper introduces a scalar tau-function that explicitly generates local conserved densities for generalized integrable hierarchies related to affine Kac-Moody algebras, simplifying the calculation of conserved charges.

Contribution

It provides a novel explicit representation of conserved densities via a single tau-function for a broad class of integrable hierarchies, linking to quantum interpretations.

Findings

Explicit tau-function representation for conserved densities

Simplified calculation of conserved charges for soliton solutions

Application to constrained KP solitons

Abstract

For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenkel and Enriquez to soliton equations of KdV and mKdV type is also discussed. These results considerably simplify the calculation of the conserved charges carried by the soliton solutions to the equations of the hierarchy, which is important to establish their interpretation as particles. By way of illustration, we calculate the charges carried by a set of constrained KP solitons recently constructed.