Tau functions and the limit of block Toeplitz determinants
Abstract
A classical way to introduce tau functions for integrable hierarchies of solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld-Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.
Keywords
Cite
@article{arxiv.1404.5149,
title = {Tau functions and the limit of block Toeplitz determinants},
author = {Mattia Cafasso and Chao-Zhong Wu},
journal= {arXiv preprint arXiv:1404.5149},
year = {2015}
}
Comments
22 pages