Gauge symmetry origin of Bäcklund transformations for Painlevé equations. (English) Zbl 1519.81295
Summary: We identify the self-similarity limit of the second flow of \(sl(N)\) mKdV hierarchy with the periodic dressing chain thus establishing a connection to \(A^{(1)}_{N-1}\) invariant Painlevé equations. The \(A^{(1)}_{N-1}\) Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra \(\widehat{sl}(N)\) endowed with a principal gradation.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
References:
| [1] | Adler V E 1993 Funct. Anal. Appl.27 141-3 · Zbl 0812.58072 · doi:10.1007/bf01085984 |
| [2] | Alves V C C 2021 On hybrid Painlevé equations Doctoral Dissertation São Paulo, Brazil-UNESP |
| [3] | Alves V C C, Aratyn H, Gomes J F and Zimerman A H 2020 J. Phys. A: Math. Theor.53 445202 · Zbl 1519.34105 · doi:10.1088/1751-8121/abb725 |
| [4] | Alves V C C, Aratyn H, Gomes J F and Zimerman A H 2019 J. Phys. A: Math. Theor.52 065203 · Zbl 1425.34107 · doi:10.1088/1751-8121/aaecdd |
| [5] | Alves V C C, Aratyn H, Gomes J F and Zimerman A H 2019 J. Phys.: Conf. Ser.1194 012002 · doi:10.1088/1742-6596/1194/1/012002 |
| [6] | Aratyn H, Gomes J F, Ruy D V and Zimerman A H 2016 J. Phys. A: Math. Theor.49 045201 · Zbl 1343.37059 · doi:10.1088/1751-8113/49/4/045201 |
| [7] | Aratyn H, Gomes J F and Zimerman A H 2011 J. Phys. A: Math. Theor.44 235202 · Zbl 1223.34121 · doi:10.1088/1751-8113/44/23/235202 |
| [8] | Aratyn H, Nissimov E, Pacheva S and Zimerman A H 1995 Int. J. Mod. Phys. A 10 2537 · Zbl 1044.37524 · doi:10.1142/s0217751x95001212 |
| [9] | Aratyn H, Ferreira L A, Gomes J F and Zimerman A H 1993 Phys. Lett. B 316 85 · doi:10.1016/0370-2693(93)90662-2 |
| [10] | Bassom A P, Clarkson P A and Hicks A C 1995 Stud. Appl. Math.95 1-71 · Zbl 0846.34006 · doi:10.1002/sapm19959511 |
| [11] | Carvalho Ferreira J M, Gomes J F, Lobo G V and Zimerman A H 2021 J. Phys. A: Math. Theor.54 065202 · Zbl 1519.37078 · doi:10.1088/1751-8121/abd8b2 |
| [12] | Cosgrove C M 2006 Stud. Appl. Math.116 321-413 · Zbl 1145.34379 · doi:10.1111/j.1467-9590.2006.00346.x |
| [13] | Flaschka H and Newell A C 1980 Commun. Math. Phys.76 65 · Zbl 0439.34005 · doi:10.1007/bf01197110 |
| [14] | Fokas A S, Mugan U and Ablowitz M J 1988 Physica D 30 247-83 · Zbl 0649.34006 · doi:10.1016/0167-2789(88)90021-8 |
| [15] | Fordy A P and Gibbons J 1980 Commun. Math. Phys.77 21-30 · Zbl 0456.35080 · doi:10.1007/bf01205037 |
| [16] | Gomes J F, Retore A L and Zimerman A H 2015 J. Phys. A: Math. Theor.48 405203 · Zbl 1325.37041 · doi:10.1088/1751-8113/48/40/405203 |
| [17] | Gomes J F, Retore A L and Zimerman A H 2016 J. Phys. A: Math. Theor.49 504003 · Zbl 1360.37165 · doi:10.1088/1751-8113/49/50/504003 |
| [18] | Gordoa P R, Joshi N and Pickering A 2005 J. Differ. Equ.217 124-53 · Zbl 1080.34073 · doi:10.1016/j.jde.2005.05.003 |
| [19] | Ince E L 1956 Ordinary Differential Equations (New York: Dover) · Zbl 0063.02971 |
| [20] | Mug̃an U and Fokas A S 1992 J. Math. Phys.33 2031-45 · Zbl 0761.34007 · doi:10.1063/1.529626 |
| [21] | Noumi M and Yamada Y 1998 Commun. Math. Phys.199 281-95 · Zbl 0952.37031 · doi:10.1007/s002200050502 |
| [22] | Noumi M and Yamada Y 1998 Funkc. Ekvacioj41 483-503 arXiv:math/9808003 · Zbl 1140.34303 |
| [23] | Schiff J 1994 Nonlinearity7 305-12 · Zbl 0838.34009 · doi:10.1088/0951-7715/7/1/015 |
| [24] | Sen A, Hone A N W and Clarkson P A 2006 Stud. Appl. Math.117 299-319 · Zbl 1145.34380 · doi:10.1111/j.1467-9590.2006.00356.x |
| [25] | Takasaki K 2003 Commun. Math. Phys.241 111-42 · Zbl 1149.81312 · doi:10.1007/s00220-003-0929-y |
| [26] | Tsuda T 2005 Adv. Math.197 587-606 · Zbl 1095.34057 · doi:10.1016/j.aim.2004.10.016 |
| [27] | Veselov A P and Shabat A B 1993 Funct. Anal. Appl.27 81-96 · Zbl 0813.35099 · doi:10.1007/bf01085979 |
| [28] | Willox R and Hietarinta J 2003 J. Phys. A: Math. Gen.36 10615-35 · Zbl 1050.34129 · doi:10.1088/0305-4470/36/42/014 |
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