Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 21 (2025), 020, 24 pages      arXiv:2306.07062      https://doi.org/10.3842/SIGMA.2025.020

Non-Integrability of the Sasano System of Type $D_5^{(1)}$ and Stokes Phenomena

Tsvetana Stoyanova
Faculty of Mathematics and Informatics, Sofia University ''St. Kliment Ohridski'', 5 J. Bourchier Blvd., Sofia 1164, Bulgaria

Received November 28, 2023, in final form March 10, 2025; Published online March 27, 2025

Abstract
In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type $D^{(1)}_l$, $l=4, 5, 6, \dots$. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group $\widetilde{W}\bigl(D^{(1)}_5\bigr)$ and which we call the Sasano system of type $D^{(1)}_5$. We prove that one family of the Sasano system of type $D^{(1)}_5$ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.

Key words: Sasano systems; non-integrability of Hamiltonian systems; differential Galois theory; Stokes phenomenon.

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References

  1. Arnol'd V.I., Mathematical methods of classical mechanics, Grad. Texts in Math., Vol. 60, Springer, New York, 1989.
  2. Ayoul M., Zung N.T., Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris 348 (2010), 1323-1326, arXiv:0901.4586.
  3. Bateman H., Erdélyi A., Higher transcendental functions. Vol. I, McGraw-Hill Book Co., Inc., New York, 1953.
  4. Boucher D., Weil J.-A., Application of J.-J. Morales and J.-P. Ramis' theorem to test the non-complete intagrability of the planar three-body problem, in From Combinatorics to Dynamical Systems, De Gruyter, Berlin, 2003, 163-178.
  5. Casale G., Morales-Ramis theorems via Malgrange pseudogroup, Ann. Inst. Fourier (Grenoble) 59 (2009), 2593-2610.
  6. Casale G., Duval G., Maciejewski A.J., Przybylska M., Integrability of Hamiltonian systems with homogeneous potentials of degree zero, Phys. Lett. A 374 (2010), 448-452, arXiv:0903.5199.
  7. Casale G., Roques J., Non-integrability by discrete quadratures, J. Reine Angew. Math. 687 (2014), 87-112.
  8. Casale G., Weil J.A., Galoisian methods for testing irreducibility of order two nonlinear differential equations, Pacific J. Math. 297 (2018), 299-337, arXiv:1504.08134.
  9. Combot T., Non-integrability of the equal mass $n$-body problem with non-zero angular momentum, Celestial Mech. Dynam. Astronom. 114 (2012), 319-340, arXiv:1112.1889.
  10. Combot T., Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues, Ann. Inst. Fourier (Grenoble) 66 (2016), 2253-2298.
  11. Combot T., Maciejewski A.J., Przybylska M., Bi-homogeneity and integrability of rational potentials, J. Differential Equations 268 (2020), 7012-7028.
  12. Filipuk G., A remark about quasi-Painlevé equations of $P_{II}$ type, C. R. Acad. Bulgare Sci. 68 (2015), 427-430.
  13. Fuji K., Suzuki T., Higher order Painlevé system of type $D^{(1)}_{2n+2}$ arising from integrable hierarchy, Int. Math. Res. Not. 2008 (2008), 129, 21 pages, arXiv:0704.2574.
  14. Loday-Richaud M., Divergent series, summability and resurgence. II. Simple and multiple summability, Lecture Notes in Math., Vol. 2154, Springer, Cham, 2016.
  15. Maciejewski A.J., Przybylska M., Gyrostatic Suslov problem, Russ. J. Nonlinear Dyn. 18 (2022), 609-627.
  16. Maciejewski A.J., Przybylska M., Simpson L., Szumiński W., Non-integrability of the dumbbell and point mass problem, Celestial Mech. Dynam. Astronom. 117 (2013), 315-330, arXiv:1304.6369.
  17. Mitschi C., Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers, Pacific J. Math. 176 (1996), 365-405.
  18. 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.
  19. Morales-Ruiz J.J., Ramis J.-P., Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal. 8 (2001), 33-96.
  20. Morales-Ruiz J.J., Ramis J.-P., Galoisian obstructions to integrability of Hamiltonian systems. II, Methods Appl. Anal. 8 (2001), 97-112.
  21. Morales-Ruiz J.J., Ramis J.-P., Integrability of dynamical systems through differential Galois theory: a practical guide, in Differential Algebra, Complex Analysis and Orthogonal Polynomials, Contemp. Math., Vol. 509, American Mathematical Society, Providence, RI, 2010, 143-220.
  22. 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. 40 (2007), 845-884.
  23. Przybylska M., Maciejewski A.J., Non-integrability of the planar elliptic restricted three-body problem, Celestial Mech. Dynam. Astronom. 135 (2023), 13, 22 pages.
  24. Ramis J.-P., Phénomène de Stokes et filtration Gevrey sur le groupe de Picard-Vessiot, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 165-167.
  25. Ramis J.-P., Gevrey asymptotics and applications to holomorphic ordinary differential equations, in Differential Equations & Asymptotic Theory in Mathematical Physics, Ser. Anal., Vol. 2, World Scientific Publishing, Hackensack, NJ, 2004, 44-99.
  26. Ramis J.-P., Epilogue: Stokes phenomena. Dynamics, classification problems and avatars, in Handbook of Geometry and Topology of Singularities VI: Foliations, Springer, Cham, 2024, 383-482.
  27. Sasano Y., Higher order Painlevé equations of type $D^{(1)}_l$, in From Soliton Theory to a Mathematics of Integrable Systems: ''New Perspectives'', RIMS Kôkyûroku Bessatsu, Vol. 1473, Res. Inst. Math. Sci. (RIMS), Kyoto, 2006, 143-163.
  28. Sasano Y., Yamada Y., Symmetry and holomorphy of Painlevé type systems, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, Vol. B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 215-225.
  29. Schlesinger L., Handbuch der Theorie der linearen Differentialgleichungen, Teubner, Leipzig, 1897.
  30. Singer M.F., Introduction to the Galois theory of linear differential equations, in Algebraic Theory of Differential Equations, London Math. Soc. Lecture Note Ser., Vol. 357, Cambridge University Press, Cambridge, 2009, 1-82, arXiv:0712.4124.
  31. Stoyanova Ts., Non-integrability of Painlevé VI equations in the Liouville sense, Nonlinearity 22 (2009), 2201-2230.
  32. Stoyanova Ts., Non-integrability of the fourth Painlevé equation in the Liouville-Arnold sense, Nonlinearity 27 (2014), 1029-1044.
  33. Stoyanova Ts., Nonintegrability of the Painlevé IV equation in the Liouville-Arnold sense and Stokes phenomena, Stud. Appl. Math. 151 (2023), 1380-1405, arXiv:2302.13732.
  34. Stoyanova Ts., Nonintegrability of coupled Painlevé systems with affine Weyl group symmetry of type $A_4^{(2)}$, Eur. J. Math. 10 (2024), 62, 15 pages.
  35. Tsygvintsev A., The meromorphic non-integrability of the three-body problem, J. Reine Angew. Math. 537 (2001), 127-149, arXiv:math.DS/0009218.
  36. Tsygvintsev A., On some exceptional cases in the integrability of the three-body problem, Celestial Mech. Dynam. Astronom. 99 (2007), 23-29, arXiv:math.DS/0610951.
  37. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren Math. Wiss., Vol. 328, Springer, Berlin, 2003.
  38. Wasow W., Asymptotic expansions for ordinary differential equations, Pure Appl. Math., Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York, 1965.

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