A Riemann--Hilbert approach to Painlevé IV

2012/07/18 by Marius van der Put, van der Put, Marius, Jaap Top +1 · 1 citation
Mathematics · Physics and Astronomy · #14D20 #14D22 #34M55 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1207.4335

openalex publication_date 2012/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlevé equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on ℙ¹ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ℙ¹ are introduced, inspired by the symmetric form for \rm PIV as was studied by M. Noumi and Y. Yamada.

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• The Instability of Painlevé Equations in Recovering Largest Eigenvalue Distributions of GUE, LUE, JUE and an Attempt of Solution to It

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