Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies

TL;DR

This paper introduces a method to compute Gelfand-Dickey tau functions via asymptotics of block Toeplitz determinants, linking integrable systems, Riemann-Hilbert problems, and algebraic geometry.

Contribution

It presents a novel approach connecting block Toeplitz determinants with Gelfand-Dickey tau functions and rational KP reductions, expanding computational tools in integrable systems.

Findings

Tau functions expressed as asymptotics of block Toeplitz determinants.

Truncated block Toeplitz determinants as tau functions for rational KP reductions.

Applications demonstrated in algebro-geometric solutions.

Abstract

We propose a method for computing any Gelfand-Dickey tau function living in Segal-Wilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols. Also truncated block Toeplitz determinants associated to the same symbols are shown to be tau function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given.