On the Linearization of the First and Second Painleve' Equations

Abstract

We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Painleve' equation with the original Garnier pair.

17 pages, 2 figures

References in corpus (3)

• Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
• Affine Weyl groups, discrete dynamical systems and Painleve equations
• On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System

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