2005/11/06 by Raimundas Vidūnas, A. V. Kitaev, Vidunas, Raimundas +1
Mathematics · Physics and Astronomy · #33E17 #34M55 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.math/0511149
openalex publication_date 2005/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences (1/2,a,b,± 1/2) and (a,a,b,b). In the present paper we give concise forms of these transformations. %, up to fractional-linear transformations. They are related to the %better known quadratic transformations obtained by Manin and Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic-pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painleve' VI functions.