### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 113, 50 pages      arXiv:1804.10369      https://doi.org/10.3842/SIGMA.2018.113
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### Three-Parameter Solutions of the PV Schlesinger-Type Equation near the Point at Infinity and the Monodromy Data

Shun Shimomura
Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Received May 01, 2018, in final form October 03, 2018; Published online October 22, 2018

Abstract
For the Schlesinger-type equation related to the fifth Painlevé equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.

Key words: Schlesinger-type equation; fifth Painlevé equation; isomonodromy deformation; monodromy data.

pdf (757 kb)   tex (54 kb)

References

1. Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1972.
2. Andreev F.V., Kitaev A.V., Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis, Nonlinearity 13 (2000), 1801-1840.
3. Bonelli G., Lisovyy O., Maruyoshi K., Sciarappa A., Tanzini A., On Painlevé/gauge theory correspondence, Lett. Math. Phys. 107 (2017), 2359-2413, arXiv:1612.06235.
4. Dubrovin B., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147, math.AG/9806056.
5. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. 1, Bateman Manuscript Project, McGraw-Hill Book Co., New York.
6. Fedoryuk M.V., Asymptotic analysis. Linear ordinary differential equations, Springer-Verlag, Berlin, 1993.
7. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
8. Gamayun O., Iorgov N., Lisovyy O., Conformal field theory of Painlevé VI, J. High Energy Phys. 2012 (2012), no. 10, 038, 25 pages, arXiv:1207.0787.
9. Gamayun O., Iorgov N., Lisovyy O., How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A: Math. Theor. 46 (2013), 335203, 29 pages, arXiv:1302.1832.
10. Gromak V.I., On the theory of Painlevé's equations, Differential Equations 11 (1975), 285-287.
11. Guzzetti D., On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation, Math. Phys. Anal. Geom. 4 (2001), 293-377, arXiv:1010.1330.
12. Guzzetti D., Tabulation of Painlevé 6 transcendents, Nonlinearity 25 (2012), 3235-3276, arXiv:1108.3401.
13. Its A.R., Novokshenov V.Yu., The isomonodromic deformation method in the theory of Painlevé equations, Lecture Notes in Math., Vol. 1191, Springer-Verlag, Berlin, 1986.
14. Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
15. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
16. Kapaev A.A., Asymptotic behavior of the solutions of the Painlevé equation of the first kind, Differential Equations 24 (1988), 1107-1115.
17. Kapaev A. A., Global asymptotics of the second Painlevé transcendent, Phys. Lett. A 167 (1992), 356-362.
18. Kapaev A.A., Global asymptotics of the fourth Painlevé transcendent, Steklov Math. Inst. and IUPUI, Preprint \# 96-06, 1996, available at http://www.pdmi.ras.ru/preprint/1996/index.html.
19. Lisovyy O., Nagoya H., Roussillon J., Irregular conformal blocks and connection formulae for Painlevé V functions, J. Math. Phys. 59 (2018), 091409, 27 pages, arXiv:1806.08344.
20. Nagoya H., Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations, J. Math. Phys. 56 (2015), 123505, 24 pages, arXiv:1505.02398.
21. Olver F.W.J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York - London, 1974.
22. Shimomura S., Series expansions of Painlevé transcendents near the point at infinity, Funkcial. Ekvac. 58 (2015), 277-319.
23. Shimomura S., The sixth Painlevé transcendents and the associated Schlesinger equation, Publ. Res. Inst. Math. Sci. 51 (2015), 417-463.
24. Shimomura S., Critical behaviours of the fifth Painlevé transcendents and the monodromy data, Kyushu J. Math. 71 (2017), 139-185, arXiv:1602.08808.
25. Sibuya Y., Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam - Oxford, 1975.
26. Takano K., A $2$-parameter family of solutions of Painlevé equation (V) near the point at infinity, Funkcial. Ekvac. 26 (1983), 79-113.
27. Wasow W., Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1965.
28. Wasow W., Linear turning point theory, Applied Mathematical Sciences, Vol. 54, Springer-Verlag, New York, 1985.