APM  Vol.1 No.4 , July 2011
Non-Integrability of Painlevé V Equations in the Liouville Sense and Stokes Phenomenon
Abstract: In this paper we are concerned with the integrability of the fifth Painlevé equation (PV ) from the point of view of the Hamiltonian dynamics. We prove that the PainlevéV equation (2) with parameters k∞=0,k0= –θ for arbitrary complex θ (and more generally with parameters related by Bäclund transformations) is non integrable by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and Ziglin-Ramis-Morales-Ruiz-Simó method yield to the non-integrable results.
Cite this paper: nullT. Stoyanova, "Non-Integrability of Painlevé V Equations in the Liouville Sense and Stokes Phenomenon," Advances in Pure Mathematics, Vol. 1 No. 4, 2011, pp. 170-183. doi: 10.4236/apm.2011.14031.

[1]   P. Painlevé, “Sur les équations Différentielles du Second Ordre et d’Ordre Supérieur dont l’Intégrale Générale est Uniforme,” Acta Mathematica, Vol. 25, No. 1, December 1902, pp. 1-86.

[2]   B. Gambier, “Sur les équations Différentielles du Second Ordre et du Premier Degré dont l’Intégrale Générale est a Points Critiques Fixes,” Acta Mathematica, Vol. 33, No. 1, December 1910, pp. 1-55. HHdoi:10.1007/BF02393211

[3]   M. Jimbo, T. Miwa, Y. Mori and M. Sato, “Density Matrix of an Impenetrable Bose Gas and the Fifth Painlevé Transcendent,” Physical D, Vol. 1, No. 1, April 1980, pp. 80-158. HHdoi:10.1016/0167-2789(80)90006-8

[4]   N. Lukashevich, “Solutions of the Fifth Equation of Painlevé,” Differential Equations, Vol. 4, 1968, pp. 732- 735.

[5]   K. Okamoto, “Studies on the Painlevé Equation . Fifth Painlevé Equatin ,” Japanese Journal of Mathematics, Vol. 13, No. 1, 1987, pp. 47-76.

[6]   J. Malmquist, “Sur Les équattions Différetielles du Second ordre dont l’Intégrale Générale a ses Points Critiques Fixes,” Astronomy Och Fysik, Vol. 18, No. 8, 1922, pp. 1-89.

[7]   J. Morales-Ruiz, “Kovalevskaya, Liapounov, Painlevé, Ziglin and Differential Galois Theory,” Regular and Chaotic Dynamics, Vol. 5, No. 3, 2000, pp. 251-272. HHdoi:10.1070/rd2000v005n03ABEH000148

[8]   J. Morales-Ruiz, “A Remark about the Painlevé Transcendents,” Théories Asymptotiques et équations de Painlevé, Sémin. Congr Angiers, Society Mathematical, Paris, Vol. 14, 2006, pp. 229-235.

[9]   Ts. Stoyanova and O. Christov, “Non-integrability of the second Painlevé Equation as a Hamiltonian System,” Compte Rendu de l’academie Bulgare des Sciences, Vol. 60, No. 1, January 2007, pp. 13-18.

[10]   E. Horozov and Ts. Stoyanova, “Non-integrability of some Painlevé VI-equations and Dilogarithms,” Regular and Chaotic Dynamics, Vol. 12, No. 6, Desember 2007, pp. 622-629.

[11]   T. Stoyanova, “Non-integrability of Painlevé VI in the Liouville Sense,” Nonlinearity, Vol. 22, No. 9, September 2009, pp. 2201-2230. HHdoi:10.1088/0951-7715/22/9/008

[12]   J. Morales-Ruiz, “Differential Galois Theory and Non- integrability of Hamiltonian Systems,” Progress in Mathematics 179, Birkh?user, Basel, 1999.

[13]   S. Ziglin, “Branching of Solutions and Non-Existence of First Integrals in Hamiltonian Mechanics I,” Functional Analysis and Its Applications, Vol. 16, No. 3, July 1982, pp. 181-189. HHdoi:10.1007/BF01081586

[14]   S. Ziglin, “Branching of Solutions and Non-existence of first Integrals in Hamiltonian Mechanics II,” Functional Analysis and Its Applications, Vol. 17, No. 1, January 1983, pp. 6-17. HHdoi:10.1007/BF01083174

[15]   J. Morales-Ruiz, J-P. Ramis and C. Simó, “Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations,” Annales Scientifiques de l’école Normale Sup’erieure, Vol. 40, No. 6, November 2007, pp. 845-884.

[16]   J. Martinet and J-P. Ramis, “Théorie de Galois différentielle et resommation,” Computer Algebra and Diffrential Equations, Academic Press, London, 1989, pp. 117-214.

[17]   J. Martinet and J-P. Ramis, “Elementary Acceleration and Multisummability,” Physique Théorique, Vol. 54, No. 4, 1991, pp. 331-401.

[18]   M. van der Put and M. Singer, “Galois Theory of Linear Differential Equations,” Grundlehren der Mathematisch- en Wissenschaflen, Springer, 2003.

[19]   C. Mitschi, “Differential Galois Groups of Confluent Generalized Hypergeometric Equations: An Approach Using Stokes Multipliers,” Pasific Journal of Mathematics, Vol. 176, No. 2, 1996, pp. 365-405.

[20]   M. Singer, “Introduction to the Galois Theory of Linear Differential Equations,” Algebraic Theory of Differential Equations, London Mathematical Society Lecture Note Series, Cambridge University Press, Vol. 2, No. 357, 2009, pp. 1-82.

[21]   J. -P. Ramis, “Séries Divergentes et Théories Asymptotiq- ues,” Bulletin Sociéte Mathématique de France, Panora- mas et Synthéses, Vol. 121, 1993, pp. 74.

[22]   J. -P. Ramis, “Gevrey Asymptotics and Applications to Holomorphic Ordinary Differential Equations,” Differential Equations and Asymptotic Theory, Seri-es in Analysis, World Scientific Singapore, Vol. 2, 2004, pp. 44-99.

[23]   W. Balser, “From Divergent Power Series to Analytic Functions,” Lecture Notes in Mathematics, Springer-Ver- lag, Berlin Heidelberg, Vol. 1582, 1994.

[24]   W. Wasov, “Asymtotic Expansions for Ordinary Differential Equations,” Dover, New York, 1965.

[25]   H. Watanabe, “Solutions of the Fifth Painlev’e Equatins I,” Hokkaido Mathematical Journal, Vol. 24, No. 2, 1995, pp. 231-268.

[26]   E. Whittaker and E. Watson, “A Course of Modern Anal- ysis,” Cambridge University Press, Cambridge, 1989.

[27]   P. Berman and M. Singer, “Calculating the Galois group of Completely Reducible Operators,” Journal of Pure and Applied Algebra, Vol. 139, No. 1-3, June 1999, pp. 3-23. HHdoi:10.1016/S0022-4049(99)00003-1

[28]   L. Schlesinger, “Handbuch der Theorie der Linearen Differentialgleichungen,” Teubner, Leipzig, 1897.

[29]   T. Masuda, Y. Ohta and K. Kajiwara, “A Determinant Formula for a Class of Rational Solutions of Painlevé V Equation,” Nagoya Mathematical Journal, Vol. 168, 2002, pp. 1-25.