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

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.

34 pages

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Cited by in corpus (4)

• From Darboux-Egorov system to bi-flat $F$-manifolds
• Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI
• Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations
• Bäcklund transformations for certain rational solutions of Painlevé VI