On the linearization of the first and second Painlevé equations

We found Fuchs-Garnier pairs in 3 X 3 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...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2009-02, Vol.42 (5), p.055208-055208 (18)
Main Authors: Joshi, N, Kitaev, A V, Treharne, P A
Format: Article
Language:English
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Summary:We found Fuchs-Garnier pairs in 3 X 3 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 of its Fuchs-Garnier pairs in 2 X 2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and that found by Harnad, Tracy and Widom. Together with the certain other transformations it allows us to relate all known 2 X 2 matrix Fuchs-Garnier pairs for the second Painleve equation to the original Garnier pair.
ISSN:1751-8121
1751-8113
1751-8121
DOI:10.1088/1751-8113/42/5/055208