Tau functions and Virasoro actions for soliton hierarchies

TL;DR

This paper develops a general framework for constructing tau functions and Virasoro actions in soliton hierarchies, providing integral formulas, exploring solution recovery, and relating tau functions to inverse scattering and Virasoro algebra actions.

Contribution

It introduces a unified method for tau function construction, integral formulas for their variations, and a general approach to Virasoro actions in soliton hierarchies.

Findings

Derived integral formulas for variations of ln(tau_f)

Established relations between tau functions and inverse scattering solutions

Constructed Virasoro algebra actions on tau functions

Abstract

There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup a solution u_f of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function tau_f for each element f in the negative subgroup. In this paper, we give integral formulas for variations of ln(tau_f) and second partials of ln(tau_f), discuss whether we can recover solutions u_f from tau_f, and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the GL(n,\C)-hierarchy.