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

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.

💡 Research Summary

The paper develops a systematic method for performing a similarity reduction of a Riemann‑Hilbert (RH) factorization problem associated with a homogeneous GL(N, ℂ) loop group and shows how this reduction yields explicit solutions of the sixth Painlevé equation (PVI). The authors begin by formulating the RH problem for the loop group: a matrix‑valued function defined on the unit circle is required to have prescribed jumps at a finite set of points and to satisfy a normalization condition at infinity. By imposing a scaling (similarity) transformation on the spectral parameter and on the loop group element, they identify a class of RH data that remain invariant under the scaling. This invariance forces the original infinite‑dimensional integrable hierarchy to collapse onto a one‑parameter family of ordinary differential equations in the scaling variable t. The compatibility condition of the associated Lax pair reproduces the standard PVI equation, establishing a direct link between the reduced hierarchy and the isomonodromic deformation problem.

A central object of the reduced hierarchy is the τ‑function, defined through the Sato‑Grassmannian construction. The authors prove that the logarithmic derivative of the τ‑function satisfies the σ‑form of PVI, i.e.
 (t(t‑1)σ″)² =