Variations for Some Painlevé Equations

This paper rst discusses irreducibility of a Painlev e equation P. We explain how the Painlev e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to a Painlev e equation P. Complete integrabili...

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Autores:: Acosta-Humañez, Primitivo B.
Van Der Put, Marius
Top, Jaap

Tipo de recurso:

Fecha de publicación:: 2019

Institución:: Universidad Simón Bolívar

Repositorio:: Repositorio Digital USB

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network_acronym_str	USIMONBOL2
network_name_str	Repositorio Digital USB
repository_id_str
dc.title.eng.fl_str_mv	Variations for Some Painlevé Equations
title	Variations for Some Painlevé Equations
spellingShingle	Variations for Some Painlevé Equations Hamiltonian systems Variational equations Painlevé equations differential Galois groups
title_short	Variations for Some Painlevé Equations
title_full	Variations for Some Painlevé Equations
title_fullStr	Variations for Some Painlevé Equations
title_full_unstemmed	Variations for Some Painlevé Equations
title_sort	Variations for Some Painlevé Equations
dc.creator.fl_str_mv	Acosta-Humañez, Primitivo B. Van Der Put, Marius Top, Jaap
dc.contributor.author.none.fl_str_mv	Acosta-Humañez, Primitivo B. Van Der Put, Marius Top, Jaap
dc.subject.eng.fl_str_mv	Hamiltonian systems Variational equations Painlevé equations differential Galois groups
topic	Hamiltonian systems Variational equations Painlevé equations differential Galois groups
description	This paper rst discusses irreducibility of a Painlev e equation P. We explain how the Painlev e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to a Painlev e equation P. Complete integrability of H is shown to imply that all solutions to P are classical (which includes algebraic), so in particular P is solvable by \quadratures". Next, we show that the variational equation of P at a given algebraic solution coincides with the normal variational equation of H at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases P2 to P5 where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the di erential Galois group for the rst two variational equations. As expected there are no cases where this group is commutative.
publishDate	2019
dc.date.accessioned.none.fl_str_mv	2019-11-12T20:08:30Z
dc.date.available.none.fl_str_mv	2019-11-12T20:08:30Z
dc.date.issued.none.fl_str_mv	2019
dc.type.eng.fl_str_mv	article
dc.type.coar.fl_str_mv	http://purl.org/coar/resource_type/c_6501
dc.identifier.issn.none.fl_str_mv	18150659
dc.identifier.uri.none.fl_str_mv	https://hdl.handle.net/20.500.12442/4330
identifier_str_mv	18150659
url	https://hdl.handle.net/20.500.12442/4330
dc.rights.*.fl_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 Internacional
dc.rights.coar.fl_str_mv	http://purl.org/coar/access_right/c_14cb
dc.rights.uri.*.fl_str_mv	http://creativecommons.org/licenses/by-nc-nd/4.0/
rights_invalid_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ http://purl.org/coar/access_right/c_14cb
dc.publisher.eng.fl_str_mv	SIGMA
dc.source.eng.fl_str_mv	Symmetry, Integrability and Geometry: Methods and Applications
dc.source.spa.fl_str_mv	Vol. 15, (2019)
institution	Universidad Simón Bolívar
dc.source.uri.eng.fl_str_mv	https://www.emis.de/journals/SIGMA/2019/088/sigma19-088.pdf
dc.source.bibliographicCitation.eng.fl_str_mv	Acosta-Humánez P.B., Nonautonomous Hamiltonian systems and Morales-Ramis theory. I. The case x = f(x; t), SIAM J. Appl. Dyn. Syst. 8 (2009), 279{297, arXiv:0808.3028. Acosta-Humánez P.B., van der Put M., Top J., Isomonodromy for the degenerate fth Painlevé equation, SIGMA 13 (2017), 029, 14 pages, arXiv:1612.03674. Casale G.,Weil J.A., Galoisian methods for testing irreducibility of order two nonlinear differential equations, Paci c J. Math. 297 (2018), 299{337, arXiv:1504.08134. Clarkson P.A., Painlevé equations - nonlinear special functions, slides presented during the IMA Summer Program Special Functions in the Digital Age, Minneapolis, July 22 - August 2, 2002, available at http: //www.math.rug.nl/~top/Clarkson.pdf. Clarkson P.A., Special polynomials associated with rational solutions of the fth Painlevé equation, J. Com- put. Appl. Math. 178 (2005), 111{129. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Editors F. Marcellán, W. Van Assche, Springer, Berlin, 2006, 331{411. Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002. Horozov E., Stoyanova T., Non-integrability of some Painlevé VI-equations and dilogarithms, Regul. Chaotic Dyn. 12 (2007), 622{629. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407{448. Lukashevich N.A., On the theory of Painlevé's third equation, Differ. Uravn. 3 (1967), 1913{1923. Lukashevich N.A., The solutions of Painlevé's fth equation, Differ. Uravn. 4 (1968), 1413{1420. Matsuda M., First-order algebraic differential equations. A differential algebraic approach, Lecture Notes in Math., Vol. 804, Springer, Berlin, 1980. Morales-Ruiz J.J., A remark about the Painlevé transcendents, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 229-235. Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001), 33{96. Morales-Ruiz J.J., Ramis J.P., Simo C., Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4) 40 (2007), 845-884. Muntingh G., van der Put M., Order one equations with the Painlevé property, Indag. Math. (N.S.) 18 (2007), 83-95, arXiv:1202.4633. Nagloo J., Pillay A., On algebraic relations between solutions of a generic Painlevé equation, J. Reine Angew. Math. 726 (2017), 1{27, arXiv:1112.2916. Ngo Chau L.X., Nguyen K.A., van der Put M., Top J., Equivalence of differential equations of order one, J. Symbolic Comput. 71 (2015), 47{59, arXiv:1303.4960. Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type PIII(D7) and PIII(D8), J. Math. Sci. Univ. Tokyo 13 (2006), 145{204. Ohyama Y., Okumura S., R. Fuchs' problem of the Painlevé equations from the rst to the fth, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 163{178, arXiv:math.CA/0512243. Stoyanova T., Non-integrability of Painlevé VI equations in the Liouville sense, Nonlinearity 22 (2009), 2201{2230. Stoyanova T., Non-integrability of Painlevé V equations in the Liouville sense and Stokes phenomenon, Adv. Pure Math. 1 (2011), 170{183. Stoyanova T., A note on the R. Fuchs's problem for the Painlevé equations, arXiv:1204.0157. Stoyanova T., Non-integrability of the fourth Painlevé equation in the Liouville-Arnold sense, Nonlinearity 27 (2014), 1029-1044. Stoyanova T., Christov O., Non-integrability of the second Painlevé equation as a Hamiltonian system, C. R. Acad. Bulgare Sci. 60 (2007), 13{18, arXiv:1103.2443. Umemura H., On the irreducibility of the rst differential equation of Painlevé, in Algebraic Geometry and Commutative Algebra, Vol. II, Kinokuniya, Tokyo, 1988, 771-789. Umemura H., Second proof of the irreducibility of the rst differential equation of Painlevé, Nagoya Math. J. 117 (1990), 125{171. Umemura H., Birational automorphism groups and differential equations, Nagoya Math. J. 119 (1990), 1{80. Umemura H., Watanabe H., Solutions of the second and fourth Painlevé equations. I, Nagoya Math. J. 148 (1997), 151{198. van der Put M., Saito M.H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611{2667, arXiv:0902.1702. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003. Z_ ol ádek H., Filipuk G., Painlevé equations, elliptic integrals and elementary functions, J. Differential Equa- tions 258 (2015), 1303{1355.
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spelling	Acosta-Humañez, Primitivo B.5f8619c8-ca55-4966-b0fc-8bdbd598be6eVan Der Put, Marius5e24ab93-6d99-4b57-873e-4eb0de06a4eaTop, Jaapff7e2b03-d42f-44c0-a0c0-bd98cd707fec2019-11-12T20:08:30Z2019-11-12T20:08:30Z201918150659https://hdl.handle.net/20.500.12442/4330This paper rst discusses irreducibility of a Painlev e equation P. We explain how the Painlev e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to a Painlev e equation P. Complete integrability of H is shown to imply that all solutions to P are classical (which includes algebraic), so in particular P is solvable by \quadratures". Next, we show that the variational equation of P at a given algebraic solution coincides with the normal variational equation of H at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases P2 to P5 where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the di erential Galois group for the rst two variational equations. As expected there are no cases where this group is commutative.SIGMAAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/http://purl.org/coar/access_right/c_14cbSymmetry, Integrability and Geometry: Methods and ApplicationsVol. 15, (2019)https://www.emis.de/journals/SIGMA/2019/088/sigma19-088.pdfAcosta-Humánez P.B., Nonautonomous Hamiltonian systems and Morales-Ramis theory. I. The case x = f(x; t), SIAM J. Appl. Dyn. Syst. 8 (2009), 279{297, arXiv:0808.3028.Acosta-Humánez P.B., van der Put M., Top J., Isomonodromy for the degenerate fth Painlevé equation, SIGMA 13 (2017), 029, 14 pages, arXiv:1612.03674.Casale G.,Weil J.A., Galoisian methods for testing irreducibility of order two nonlinear differential equations, Paci c J. Math. 297 (2018), 299{337, arXiv:1504.08134.Clarkson P.A., Painlevé equations - nonlinear special functions, slides presented during the IMA Summer Program Special Functions in the Digital Age, Minneapolis, July 22 - August 2, 2002, available at http: //www.math.rug.nl/~top/Clarkson.pdf.Clarkson P.A., Special polynomials associated with rational solutions of the fth Painlevé equation, J. Com- put. Appl. Math. 178 (2005), 111{129.Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Editors F. Marcellán, W. Van Assche, Springer, Berlin, 2006, 331{411.Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.Horozov E., Stoyanova T., Non-integrability of some Painlevé VI-equations and dilogarithms, Regul. Chaotic Dyn. 12 (2007), 622{629.Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407{448.Lukashevich N.A., On the theory of Painlevé's third equation, Differ. Uravn. 3 (1967), 1913{1923.Lukashevich N.A., The solutions of Painlevé's fth equation, Differ. Uravn. 4 (1968), 1413{1420.Matsuda M., First-order algebraic differential equations. A differential algebraic approach, Lecture Notes in Math., Vol. 804, Springer, Berlin, 1980.Morales-Ruiz J.J., A remark about the Painlevé transcendents, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 229-235.Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001), 33{96.Morales-Ruiz J.J., Ramis J.P., Simo C., Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4) 40 (2007), 845-884.Muntingh G., van der Put M., Order one equations with the Painlevé property, Indag. Math. (N.S.) 18 (2007), 83-95, arXiv:1202.4633.Nagloo J., Pillay A., On algebraic relations between solutions of a generic Painlevé equation, J. Reine Angew. Math. 726 (2017), 1{27, arXiv:1112.2916.Ngo Chau L.X., Nguyen K.A., van der Put M., Top J., Equivalence of differential equations of order one, J. Symbolic Comput. 71 (2015), 47{59, arXiv:1303.4960.Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type PIII(D7) and PIII(D8), J. Math. Sci. Univ. Tokyo 13 (2006), 145{204.Ohyama Y., Okumura S., R. Fuchs' problem of the Painlevé equations from the rst to the fth, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 163{178, arXiv:math.CA/0512243.Stoyanova T., Non-integrability of Painlevé VI equations in the Liouville sense, Nonlinearity 22 (2009), 2201{2230.Stoyanova T., Non-integrability of Painlevé V equations in the Liouville sense and Stokes phenomenon, Adv. Pure Math. 1 (2011), 170{183.Stoyanova T., A note on the R. Fuchs's problem for the Painlevé equations, arXiv:1204.0157.Stoyanova T., Non-integrability of the fourth Painlevé equation in the Liouville-Arnold sense, Nonlinearity 27 (2014), 1029-1044.Stoyanova T., Christov O., Non-integrability of the second Painlevé equation as a Hamiltonian system, C. R. Acad. Bulgare Sci. 60 (2007), 13{18, arXiv:1103.2443.Umemura H., On the irreducibility of the rst differential equation of Painlevé, in Algebraic Geometry and Commutative Algebra, Vol. II, Kinokuniya, Tokyo, 1988, 771-789.Umemura H., Second proof of the irreducibility of the rst differential equation of Painlevé, Nagoya Math. J. 117 (1990), 125{171.Umemura H., Birational automorphism groups and differential equations, Nagoya Math. J. 119 (1990), 1{80.Umemura H., Watanabe H., Solutions of the second and fourth Painlevé equations. I, Nagoya Math. J. 148 (1997), 151{198.van der Put M., Saito M.H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611{2667, arXiv:0902.1702.van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.Z_ ol ádek H., Filipuk G., Painlevé equations, elliptic integrals and elementary functions, J. 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Variations for Some Painlevé Equations

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