On the properties of Bäcklund transformations on Painleväé 6th equations

Authors

  • Amarzaya Amartuvshin Department of Mathematics, School of Arts and Sciences, National University of Mongolia, Ulaanbaatar, Mongolia
  • Davaadorj Balbaan Department of Mathematics, School of Arts and Sciences, National University of Mongolia, Ulaanbaatar, Mongolia

DOI:

https://doi.org/10.5564/jasea.v4i1.3222

Keywords:

Painlevé equations , Bäcklund transformations, Affine Weyl group, Bonnet surfaces

Abstract

In his series of papers Okamoto give detailed explanation of Painlevé equations of types 2-6. On the other hand Kajiwara et.al described Bäcklund transformations for these types of Painlevé equations. Later, Bobenko and Eitner found that Painlevé equations of types 6 and 5 give rise Bonnet surfaces and vise-versa. In this research we study properties of Bäcklund transformations of 6th Painlevé equations P6 and determine explicit formula of such transformations which give a new Bonnet surfaces.

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References

A. Bobenko and U.Eitner, Painlevé equations in the differential geometry, Lecture notes in mathematics, 1753, (2000). https://doi.org/10.1007/b76883

K. Kajiwara, T.Masuda, M.Noumi, Y.Ohta and Y.Yamada, Determinant formulas for the Toda and discrete Toda equations, Funkcial. Ekvac. 44 (2001), pp. 291–307.

T.Masuda, Classical transcendental solutions of the Painlevé equations and their degeneration, Tohoku Math. J. 56 (2004), pp. 467–490. https://doi.org/10.2748/tmj/1113246745

M.Noumi, Painlevé equations through symmetry, Translations of mathematical monographs, 2004. https://doi.org/10.1090/mmono/223

K.Okamoto, Studies on the Painlevé equations I, sixth Painlevé equation PVI, Ann. Mat. Pura Appl. 4- 146 (1987), pp. 337–381. https://doi.org/10.1007/BF01762370

K.Okamoto, Studies on the Painlevé equations II, fifth Painlevé equation PV, Japan J. Math. 13 (1987), pp. 47–76. https://doi.org/10.4099/math1924.13.47

K.Okamoto, Studies on the Painlevé equations III, second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), pp. 222–254. https://doi.org/10.1007/BF01458459

K.Okamoto, Studies on the Painlevé equations IV, third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), pp. 305–332.

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Published

2023-12-28

How to Cite

[1]
A. Amartuvshin and D. Balbaan, “On the properties of Bäcklund transformations on Painleväé 6th equations”, J. appl. sci. eng., A, vol. 4, no. 1, pp. 8–15, Dec. 2023.

Issue

Vol. 4 No. 1 (2023)

Section

Articles

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Copyright (c) 2023 Amarzaya Amartuvshin, Davaadorj Balbaan

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