Solutions of the Painleve VI Equation from Reduction of Integrable Hierarchy in a Grassmannian Approach

TL;DR

This paper develops a Grassmannian-based method to explicitly reduce an integrable hierarchy, leading to rational solutions of the Painleve VI equation through a Riemann-Hilbert problem approach.

Contribution

It introduces a novel Grassmannian formulation for constructing tau functions that yield explicit rational solutions to Painleve VI.

Findings

Derived explicit solutions to Painleve VI from integrable hierarchy reduction.

Established a Grassmannian framework for tau function construction.

Connected Riemann-Hilbert problems with Painleve VI solutions.

Abstract

We present an explicit method to perform similarity reduction of a Riemann-Hilbert factorization problem for a homogeneous GL (N, C) loop group and use our results to find solutions to the Painleve VI equation for N=3. The tau function of the reduced hierarchy is shown to satisfy the sigma-form of the Painleve VI equation. A class of tau functions of the reduced integrable hierarchy is constructed by means of a Grassmannian formulation. These solutions provide rational solutions of the Painleve VI equation.