Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

Autor: Primitivo B. Acosta-Humánez, Chara Pantazi

Publikováno v: Symmetry, Integrability and Geometry: Methods and Applications, Vol 8, p 043 (2012)

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the ''invariance'' of the objects of the ''Darboux theory of integrability''. In parti

Externí odkaz: https://doaj.org/article/4df47e1e16cb41adb1a20d5c8fdfcbbc

Sextic anharmonic oscillators and Heun differential equations

Autor: Primitivo B. Acosta-Humánez, Mourad E. H. Ismail, Nasser Saad

Publikováno v: The European Physical Journal Plus. 137

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_________::fe4b230ba837695cfdb6f389cfbf720a
https://doi.org/10.1140/epjp/s13360-022-03029-3

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Autor: Primitivo B. Acosta Humánez, Jorge Rodríguez Contreras, Alberto Reyes Linero, Bladimir Blanco Montes

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critica

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::b6b1c67b2d638b5d99ae0df97219624f
http://arxiv.org/abs/2103.02773

Nonintegrability of the Armbruster--Guckenheimer--Kim Quartic Hamiltonian Through Morales--Ramis Theory

Autor: Martha Alvarez-Ramírez, Primitivo B. Acosta-Humánez, T. J. Stuchi

Publikováno v: Vol. 17, No.1 (2018)
Revista SIAM
Repositorio Digital USB
Universidad Simón Bolívar
instacron:Universidad Simón Bolívar

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincar\'e sections to illustrate the breakdown of

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::c0e2438473adfccb20c0693a9112bcb8
https://doi.org/10.1137/16m1080689

Variations for some Painlevé Equations

Autor: Jaap Top, Marius van der Put, Primitivo B. Acosta-Humánez

Publikováno v: Symmetry, Integrability and Geometry, 15:088. NATL ACAD SCI UKRAINE, INST MATH
Symmetry, Integrability and Geometry: Methods and Applications
Vol. 15, (2019)
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.

This paper first 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 Hamil

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::d97ef2df4f95eae41c08714581034df3
https://doi.org/10.3842/SIGMA.2019.088