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

SIGMA 21 (2025), 097, 50 pages      arXiv:2411.08853      https://doi.org/10.3842/SIGMA.2025.097

Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations

Malik Balogoun and Marco Bertola
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, H3G 1M8 Canada

Received November 14, 2024, in final form October 28, 2025; Published online November 14, 2025

Abstract
In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.

Key words: Painlevé equations; rational solutions; asymptotic analysis; Riemann-Hilbert problems.

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References

  1. Aratyn H., Gomes J.F., Lobo G.V., Zimerman A.H., Why is my rational Painlevé V solution not unique?, arXiv:2307.07825.
  2. Aratyn H., Gomes J.F., Lobo G.V., Zimerman A.H., Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions, Mathematics 12 (2024), 3701, 26 pages, arXiv:2409.03534.
  3. Aratyn H., Gomes J.F., Lobo G.V., Zimerman A.H., Two-fold degeneracy of a class of rational Painlevé V solutions, Open Commun. Nonlinear Math. Phys. (2024), spec. issue 2, 28-45, arXiv:2310.01585.
  4. Balogh F., Bertola M., Bothner T., Hankel determinant approach to generalized Vorob'ev-Yablonski polynomials and their roots, Constr. Approx. 44 (2016), 417-453, arXiv:1504.00440.
  5. Barhoumi A., Celsus A.F., Deaño A., Global-phase portrait and large-degree asymptotics for the Kissing polynomials, Stud. Appl. Math. 147 (2021), 448-526, arXiv:2008.08724.
  6. Bertola M., Bothner T., Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity, Int. Math. Res. Not. 2015 (2015), 9330-9399, arXiv:1401.1408.
  7. Bertola M., Eynard B., Harnad J., Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions, Comm. Math. Phys. 263 (2006), 401-437, arXiv:nlin.SI/0410043.
  8. Bertola M., Katsevich A., Tovbis A., Singular value decomposition of a finite Hilbert transform defined on several intervals and the interior problem of tomography: the Riemann-Hilbert problem approach, Comm. Pure Appl. Math. 69 (2016), 407-477, arXiv:1402.0216.
  9. Bertola M., Lee S.Y., First colonization of a hard-edge in random matrix theory, Constr. Approx. 31 (2010), 231-257, arXiv:0804.1111.
  10. Bertola M., Mo M.Y., Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights, Adv. Math. 220 (2009), 154-218, arXiv:math-ph/0605043.
  11. Bleher P., Its A., Double scaling limit in the random matrix model: the Riemann-Hilbert approach, Comm. Pure Appl. Math. 56 (2003), 433-516, arXiv:math-ph/0201003.
  12. Borodin A., Forrester P.J., Increasing subsequences and the hard-to-soft edge transition in matrix ensembles, J. Phys. A 36 (2003), 2963-2981, arXiv:math-ph/0205007.
  13. Bothner T., Miller P.D., Rational solutions of the Painlevé-III equation: large parameter asymptotics, Constr. Approx. 51 (2020), 123-224, arXiv:1808.01421.
  14. Bothner T., Miller P.D., Sheng Y., Rational solutions of the Painlevé-III equation, Stud. Appl. Math. 141 (2018), 626-679, arXiv:1801.04360.
  15. Buckingham R.J., Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials, Int. Math. Res. Not. 2020 (2020), 5534-5577, arXiv:1706.09005.
  16. Buckingham R.J., Miller P.D., The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix, J. Anal. Math. 118 (2012), 397-492, arXiv:1106.5716.
  17. Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour, Nonlinearity 27 (2014), 2489-2578, arXiv:1310.2276.
  18. Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: critical behaviour, Nonlinearity 28 (2015), 1539-1596, arXiv:1406.0826.
  19. Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-IV solutions by the isomonodromy method, Constr. Approx. 56 (2022), 233-443, arXiv:2008.00600.
  20. Clarkson P.A., The fourth Painlevé equation and associated special polynomials, J. Math. Phys. 44 (2003), 5350-5374.
  21. Clarkson P.A., The third Painlevé equation and associated special polynomials, J. Phys. A 36 (2003), 9507-9532.
  22. Clarkson P.A., Special polynomials associated with rational solutions of the Painlevé equations and applications to soliton equations, Comput. Methods Funct. Theory 6 (2006), 329-401, arXiv:nlin.SI/0605034.
  23. Clarkson P.A., Dunning C., Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials, Stud. Appl. Math. 152 (2024), 453-507, arXiv:2304:01579.
  24. Clarkson P.A., Mansfield E.L., The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16 (2003), R1-R26.
  25. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, 1999.
  26. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335-1425.
  27. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368.
  28. Fokas A.S., Its A.R., Kitaev A.V., Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313-344.
  29. Gakhov F.D., Boundary value problems, Dover Publications, Inc., New York, 1990.
  30. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1-55.
  31. Hastings S.P., McLeod J.B., A boundary value problem associated with the second Painlevé transcendent and the Korteweg-dethinspace Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31-51.
  32. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  33. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  34. Joshi N., Kajiwara K., Mazzocco M., Generating function associated with the determinant formula for the solutions of the Painlevé II equation, Astérisque 297 (2004), 67-78, arXiv:nlin.SI/0406035.
  35. Kajiwara K., Ohta Y., Determinant structure of the rational solutions for the Painlevé II equation, J. Math. Phys. 37 (1996), 4693-4704, arXiv:solv-int/9607002.
  36. Kitaev A.V., Law C.K., McLeod J.B., Rational solutions of the fifth Painlevé equation, Differential Integral Equations 7 (1994), 967-1000.
  37. Marcellán F., Rocha I.A., On semiclassical linear functionals: integral representations, J. Comput. Appl. Math. 57 (1995), 239-249.
  38. Marcellán F., Rocha I.A., Complex path integral representation for semiclassical linear functionals, J. Approx. Theory 94 (1998), 107-127.
  39. Maroni P., Prolégomènes à l'étude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. 149 (1987), 165-184.
  40. Masoero D., Roffelsen P., Poles of Painlevé IV rationals and their distribution, SIGMA 14 (2018), 002, 49 pages, arXiv:1707.05222.
  41. Masoero D., Roffelsen P., Roots of generalised Hermite polynomials when both parameters are large, Nonlinearity 34 (2021), 1663-1732, arXiv:1907.08552.
  42. Masuda T., Some properties of zeros of the umemura polynomials of the fifth Painlevé equation, Stud. Appl. Math. 155 (2025), e70109, 34 pages.
  43. Masuda T., Ohta Y., Kajiwara K., A determinant formula for a class of rational solutions of Painlevé V equation, Nagoya Math. J. 168 (2002), 1-25, arXiv:nlin.SI/0101056.
  44. Mazzocco M., Rational solutions of the Painlevé VI equation, J. Phys. A 34 (2001), 2281-2294, arXiv:nlin.SI/0007036.
  45. McCoy B.M., Tracy C.A., Wu T.T., Connection between the KdV equation and the two-dimensional Ising model, Phys. Lett. A 61 (1977), 283-284.
  46. McCoy B.M., Tracy C.A., Wu T.T., Painlevé functions of the third kind, J. Math. Phys. 18 (1977), 1058-1092.
  47. Miller P.D., Sheng Y., Rational solutions of the Painlevé-II equation revisited, SIGMA 13 (2017), 065, 29 pages, arXiv:1704.04851.
  48. Noumi M., Yamada Y., Umemura polynomials for the Painlevé V equation, Phys. Lett. A 247 (1998), 65-69.
  49. Novokshenov V.Yu., Shchelkonogov A.A., Double scaling limit in the Painlevé IV equation and asymptotics of the Okamoto polynomials, in Spectral Theory and Differential Equations, Amer. Math. Soc. Transl. Ser. 2, Vol. 233, American Mathematical Society, Providence, RI, 2014, 199-210.
  50. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{{\rm II}}$ and $P_{{\rm IV}}$, Math. Ann. 275 (1986), 221-255.
  51. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\rm V}$, Japan. J. Math. (N.S.) 13 (1987), 47-76.
  52. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation $P_{{\rm III}}$, Funkcial. Ekvac. 30 (1987), 305-332.
  53. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A. (Editors), NIST Digital Library of Mathematical Functions, 2025, Release 1.2.4.
  54. Painlevé P., Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), 1-85.
  55. Shohat J., A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), 401-417.
  56. Strebel K., Quadratic differentials, Ergeb. Math. Grenzgeb. (3), Vol. 5, Springer, Berlin, 1984.
  57. Tracy C.A., Widom H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174, arXiv:hep-th/9211141.
  58. Umemura H., Special polynomials associated with the Painlevé equations I, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), 1063-1089.
  59. Vanlessen M., Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx. 25 (2007), 125-175, arXiv:math.CA/0504604.
  60. Vorob'ev A.P., On the rational solutions of the second Painlevé equation, Differ. Equ. 1 (1965), 79-81.
  61. Yablonskii A.I., Residues of poles of solutions of Painlevé's second equation, Dokl. Akad. Nauk BSSR 4 (1960), 47-50.

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