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.
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https://hal.archives-ouvertes.fr/hal-00989747
Contributeur : Mattia Cafasso <>
Soumis le : lundi 12 mai 2014 - 13:35:43
Dernière modification le : lundi 5 février 2018 - 15:00:03

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  • HAL Id : hal-00989747, version 1
  • ARXIV : 1404.5149

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Mattia Cafasso, Chao-Zhong Wu. Tau functions and the limit of block Toeplitz determinants. Int. Math. Res. Not. IMRN, 2015. 〈hal-00989747〉

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