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Cubic and quartic transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence. (English) Zbl 1281.34132

Local solutions of the sixth Painlevé equation (PVI) are in one-to-one correspondence (Riemann-Hilbert correspondence) with the points on the monodromy manifold associated to the Fuchsian system, which is a certain cubic surface in \(\mathbb{C}^3\). The authors revisit quadratic and quartic transformations of PVI and study the corresponding actions on the monodromy manifold, which are given by quadratic polynomial transformations of the cubic surface. The authors also classify all quadratic polynomial transformations of the cubic surface and find a new cubic transformation of the Picard case of PVI.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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[1] Fuchs, Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären, Stellen. Math. Ann. 63 pp 301– (1907) · JFM 38.0362.01 · doi:10.1007/BF01449199
[2] Schlesinger, Uber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten, J. fur Math 141 pp 96– (1912)
[3] Garnier, Solution du probleme de Riemann pour les systemes différentielles linéaires du second ordre, Ann. Sci. Ecole Norm. Sup. 43 pp 239– (1926)
[4] Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. RIMS, Kyoto Univ. 18 pp 1137– (1982) · Zbl 0535.34042 · doi:10.2977/prims/1195183300
[5] Dubrovin, Monodromy of certain Painlevé-VI transcendents and reflection group, Invent. Math. 141 pp 55– (2000) · Zbl 0960.34075 · doi:10.1007/PL00005790
[6] Iwasaki, An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys. 242 pp 185– (2003) · Zbl 1044.34051
[7] Kitaev, Quadratic transformations for the sixth Painlevé equation, Lett. Math. Phys. 21 pp 105– (1991) · Zbl 0716.34075 · doi:10.1007/BF00401643
[8] Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of : Geometry of Differential Equations, Amer. Math. Soc. Transl. Ser. 186(2) pp 131– (1998)
[9] Ramani, Quadratic relations in continous and discrete Painlevé equations, J. Phys. A. Math. Gen. 33 pp 3033– (2000) · Zbl 0953.34078 · doi:10.1088/0305-4470/33/15/310
[10] Tsuda, Folding transformations of the Painlevé equations, Math. Ann. 331 pp 713– (2005) · Zbl 1073.34101 · doi:10.1007/s00208-004-0600-8
[11] Filipuk, On the transformations of the sixth Painlevé equation, J. Nonlinear Math. Phys. 10 pp 57– (2003)
[12] Chekhov, Isomonodromic deformations and twisted Yangians arising in Teichmüller theory, Adv. Math. 226 pp 4731– (2010) · Zbl 1216.32008 · doi:10.1016/j.aim.2010.12.017
[13] Ugaglia, On a Poisson structure on the space of Stokes matrices, Int. Math. Res. Not. pp 473– (1999) · Zbl 0939.34073 · doi:10.1155/S1073792899000240
[14] Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 pp 30– (1995) · Zbl 0861.53049
[15] Silverman, The arithmetic of elliptic curves. (2nd ed.), Graduate Texts in Mathematics 106 (2009) · Zbl 1194.11005
[16] Etingof, Noncommutative del Pezzo surfaces and Calabi-Yau algebras, J. Eur. Math. Soc. 12 (6) pp 1371– (2010) · Zbl 1204.14004 · doi:10.4171/JEMS/235
[17] Vidunas, Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions, Mathematische Nachrichten 280 pp 1834– (2007) · Zbl 1137.34041 · doi:10.1002/mana.200510582
[18] Inaba, Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, IMRN 2004 pp 1– (2004) · Zbl 1087.34062 · doi:10.1155/S1073792804131310
[19] Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 pp 157– (2001) · doi:10.1007/PL00004500
[20] Galbraith, Mathematics of public key cryptography (2012) · Zbl 1238.94027 · doi:10.1017/CBO9781139012843
[21] Jimbo, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, Physica 2D 2 pp 306– (1981)
[22] Jimbo, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II, Physica 2D pp 2407– (1981) · Zbl 1194.34166
[23] Bolibruch, Developments in Mathematics: the Moscow School pp 54– ((1993))
[24] Okamoto, Polynomial Hamiltonians associated with Painlevé equations. I, Proc. Japan Acad. Ser. A Math. Sci. 56 pp 264– (1980) · Zbl 0476.34010 · doi:10.3792/pjaa.56.264
[25] Okamoto, Studies on the Painlev equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. 146(4) pp 337– (1987)
[26] Noumi, Microlocal Analysis and Complex Fourier Analysis pp 238– (2002) · doi:10.1142/9789812776594_0016
[27] O. Lisovyy YU. Tykhyy Algebraic solutions of the sixth Painlevé equation arXiv:0809.4873 2008
[28] Dubrovin, Canonical structure and symmetries of the Schlesinger equations, Comm. Math. Phys. 271 pp 289– (2007) · Zbl 1146.32005 · doi:10.1007/s00220-006-0165-3
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