On the linearization of the first and second Painlevé equations

N. Joshi, A. V. Kitaev and P. A. Treharne

Abstract

We find Fuchs–Garnier pairs in the form of 3×3 matrices for the first and second Painlevé equations which are linear in the spectral parameter. As an application of our pairs for the second Painlevé equation, we use the generalized Laplace transform to derive an invertible integral transformation relating two Fuchs–Garnier pairs in 2×2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with certain other transformations, it allows us to relate all known 2×2 matrix Fuchs–Garnier pairs for the second Painlevé equation with the original Garnier pair.

Keywords: Painlevé Equations, Isomonodromy Deformations, Laplace Transform, Lax Pair, Stokes Phenomenon.

AMS Subject Classification: Primary 33E17; secondary 34M25, 34M55.