On the Linearization of the First and Second Painlevé Equations
نویسندگان
چکیده
We found Fuchs–Garnier pairs in 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 its 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 the 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. 2000 Mathematics Subject Classification: 33E17, 34M25, 34M55. Short title: Linearization of Painlevé I and II Equations
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