New transformations for Painlevé’s third transcendent
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Abstract:
We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system $B^{(1)}_2$, to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the $\tau$-functions with these parameter sets.References
- H. Airault, Rational solutions of Painlevé equations, Stud. Appl. Math. 61 (1979), no. 1, 31–53. MR 535866, DOI 10.1002/sapm197961131
- L. A. Bordag and A. V. Kitaev, Preobrazovaniya resheniĭ tret′ego i pyatogo uravneniĭ Penleve i ikh chastnye resheniya, Soobshcheniya Ob″edinennogo Instituta Yadernykh Issledovaniĭ . Dubna [Communications of the Joint Institute for Nuclear Research. Dubna], R5-85-740, Joint Inst. Nuclear Res., Dubna, 1985 (Russian). With an English summary. MR 831426
- A. S. Fokas and M. J. Ablowitz, On a unified approach to transformations and elementary solutions of Painlevé equations, J. Math. Phys. 23 (1982), no. 11, 2033–2042. MR 679998, DOI 10.1063/1.525260
- P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys. 219 (2001), no. 2, 357–398. MR 1833807, DOI 10.1007/s002200100422
- B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l‘intégrale générale est a points critiques fixes, Acta Math. 33 (1909), 1–55.
- P. R. Gordoa, N. Joshi, and A. Pickering, Mappings preserving locations of movable poles. II. The third and fifth Painlevé equations, Nonlinearity 14 (2001), no. 3, 567–582. MR 1830907, DOI 10.1088/0951-7715/14/3/307
- P. R. Gordoa, N. Joshi, and A. Pickering, Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations, Glasg. Math. J. 43A (2001), 23–32. Integrable systems: linear and nonlinear dynamics (Islay, 1999). MR 1869683, DOI 10.1017/S0017089501000039
- V. I. Gromak, The solutions of Painlevé’s third equation, Differencial′nye Uravnenija 9 (1973), 2082–2083, 2118 (Russian). MR 0340683
- V. I. Gromak, On the theory of Painlevé’s equations, Differencial′nye Uravnenija 11 (1975), 373–376, 398 (Russian). MR 0377148
- V. I. Gromak, The solutions of Painlevé’s fifth equation, Differencial′nye Uravnenija 12 (1976), no. 4, 740–742, 775 (Russian). MR 0430372
- V. I. Gromak, Reducibility of the Painlevé equations, Differentsial′nye Uravneniya 20 (1984), no. 10, 1674–1683 (Russian). MR 767875
- Valerii I. Gromak, Bäcklund transformations of Painlevé equations and their applications, The Painlevé property, CRM Ser. Math. Phys., Springer, New York, 1999, pp. 687–734. MR 1714703
- V. I. Gromak and G. V. Filipuk, On functional relations between solutions of the fifth Painlevé equation, Differ. Uravn. 37 (2001), no. 5, 586–591, 717 (Russian, with Russian summary); English transl., Differ. Equ. 37 (2001), no. 5, 614–620. MR 1850721, DOI 10.1023/A:1019204329101
- Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, and Yasuhiko Yamada, Determinant formulas for the Toda and discrete Toda equations, Funkcial. Ekvac. 44 (2001), no. 2, 291–307. MR 1865393
- N. A. Lukaševič, On the theory of Painlevé’s third equation, Differencial′nye Uravnenija 3 (1967), 1913–1923 (Russian). MR 0229885
- Elizabeth L. Mansfield and Helen N. Webster, On one-parameter families of Painlevé III, Stud. Appl. Math. 101 (1998), no. 3, 321–341. MR 1645298, DOI 10.1111/1467-9590.00096
- Alice E. Milne, Peter A. Clarkson, and Andrew P. Bassom, Bäcklund transformations and solution hierarchies for the third Painlevé equation, Stud. Appl. Math. 98 (1997), no. 2, 139–194. MR 1431877, DOI 10.1111/1467-9590.00044
- Yoshihiro Murata, Classical solutions of the third Painlevé equation, Nagoya Math. J. 139 (1995), 37–65. MR 1355268, DOI 10.1017/S0027763000005298
- M. Noumi, Painlevé equations: An introduction from the symmetric point of view, Asakura Shoten Publishing, Tokyo, 2000, in Japanese.
- Kazuo Okamoto, Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{\textrm {II}}$ and $P_{\textrm {IV}}$, Math. Ann. 275 (1986), no. 2, 221–255. MR 854008, DOI 10.1007/BF01458459
- Kazuo Okamoto, Studies on the Painlevé equations. IV. Third Painlevé equation $P_{\textrm {III}}$, Funkcial. Ekvac. 30 (1987), no. 2-3, 305–332. MR 927186
- P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l‘intégrale générale est uniforme, Acta Math. 25 (1902), 1–85.
- Hiroshi Umemura and Humihiko Watanabe, Solutions of the third Painlevé equation. I, Nagoya Math. J. 151 (1998), 1–24. MR 1650348, DOI 10.1017/S0027763000025149
- N. S. Witte, Gap probabilities for double intervals in Hermitian random matrix ensembles as $\tau$-functions - the Bessel kernel case, in preparation, 2001.
Additional Information
- N. S. Witte
- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
- Email: N.Witte@ms.unimelb.edu.au
- Received by editor(s): January 26, 2002
- Received by editor(s) in revised form: June 1, 2002
- Published electronically: January 27, 2004
- Communicated by: Mark J. Ablowitz
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1649-1658
- MSC (2000): Primary 34M55, 33E17; Secondary 20F55
- DOI: https://doi.org/10.1090/S0002-9939-04-07087-X
- MathSciNet review: 2051125