Rational Solutions of the Painlevé-II Equation Revisited

Peter D. Miller ; Yue Sheng

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann–Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann–Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka–Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo–Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka–Newell and Bertola–Bothner Riemann–Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann–Hilbert representations by means of the steepest descent method.

Détail

Keywords: Painlevé equations; rational functions; Riemann–Hilbert problems; steepest descent method.

Bibliographie

[1] Airault H., “Rational solutions of Painlevé equations”, Stud. Appl. Math., 61 (1979), 31–53 | DOI | MR | Zbl

[2] Baik J., Deift P., Johansson K., “On the distribution of the length of the longest increasing subsequence of random permutations”, J. Amer. Math. Soc., 12 (1999), 1119–1178, arXiv: math.CO/9810105 | DOI | MR | Zbl

[3] Bass L., “Electrical structures of interfaces in steady electrolysis”, Trans. Faraday Soc., 60 (1964), 1656–1663 | DOI

[4] Bass L., Nimmo J. J. C., Rogers C., Schief W. K., “Electrical structures of interfaces: a Painlevé II model”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117–2136 | DOI | MR | Zbl

[5] 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 | DOI | MR | Zbl

[6] Bertola M., Cafasso M., “Darboux transformations and random point processes”, Int. Math. Res. Not., 2015 (2015), 6211–6266, arXiv: 1401.4752 | DOI | MR | Zbl

[7] 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 | DOI | MR | Zbl

[8] Buckingham R. J., Miller P. D., “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour”, Nonlinearity, 27 (2014), 2489–2578, arXiv: 1310.2276 | DOI | MR

[9] Buckingham R. J., Miller P. D., “Large-degree asymptotics of rational Painlevé-II functions: critical behaviour”, Nonlinearity, 28 (2015), 1539–1596, arXiv: 1406.0826 | DOI | MR | Zbl

[10] 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 | DOI | MR | Zbl

[11] Clarkson P. A., “Vortices and polynomials”, Stud. Appl. Math., 123 (2009), 37–62, arXiv: 0901.0139 | DOI | MR | Zbl

[12] Clarkson P. A., Mansfield E. L., “The second {P}ainlevé equation, its hierarchy and associated special polynomials”, Nonlinearity, 16 (2003), R1–R26 | DOI | MR | Zbl

[13] Costin O., Huang M., Tanveer S., “Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_I$”, Duke Math. J., 163 (2014), 665–704, arXiv: 1209.1009 | DOI | MR | Zbl

[14] Deift P., Kriecherbauer T., McLaughlin K. T.-R., Venakides S., Zhou X., “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure Appl. Math., 52 (1999), 1491–1552 | 3.3.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[15] 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 | DOI | MR | Zbl

[16] Flaschka H., Newell A. C., “Monodromy- and spectrum-preserving deformations. I”, Comm. Math. Phys., 76 (1980), 65–116 | DOI | MR | Zbl

[17] Fokas A. S., Grammaticos B., Ramani A., “From continuous to discrete Painlevé equations”, J. Math. Anal. Appl., 180 (1993), 342–360 | DOI | MR | Zbl

[18] 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 | DOI | MR | Zbl

[19] Fornberg B., Weideman J. A. C., “A computational exploration of the second Painlevé equation”, Found. Comput. Math., 14 (2014), 985–1016 | DOI | MR | Zbl

[20] Fukutani S., Okamoto K., Umemura H., “Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations”, Nagoya Math. J., 159 (2000), 179–200 | DOI | MR | Zbl

[21] Iwasaki K., Kajiwara K., Nakamura T., “Generating function associated with the rational solutions of the Painlevé II equation”, J. Phys. A: Math. Gen., 35 (2002), L207–L211, arXiv: nlin.SI/0112043 | DOI | MR | Zbl

[22] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, E16, Friedr. Vieweg Sohn, Braunschweig, 1991 | DOI | MR

[23] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D, 2 (1981), 407–448 | DOI | MR | Zbl

[24] Johnson C. V., String theory without branes, arXiv: hep-th/0610223

[25] Joshi N., Kitaev A. V., Treharne P. A., “On the linearization of the first and second Painlevé equations”, J. Phys. A: Math. Gen., 42 (2009), 055208, 18 pp., arXiv: 0806.0271 | DOI | MR | Zbl

[26] 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 | DOI | MR | Zbl

[27] Kametaka Y., Noda M., Fukui Y., Hirano S., “A numerical approach to Toda equation and Painlevé II equation”, Mem. Fac. Eng. Ehime Univ., 9 (1986), 1–24

[28] Kapaev A. A., “Scaling limits in the second Painlevé transcendent”, J. Math. Sci., 83 (1997), 38–61 | DOI | MR

[29] Kapaev A. A., “Quasi-linear stokes phenomenon for the Painlevé first equation”, J. Phys. A: Math. Gen., 37 (2004), 11149–11167, arXiv: nlin.SI/0404026 | DOI | MR | Zbl

[30] Kapaev A. A., Kitaev A. V., “Passage to the limit ${\rm P}_2\to{\rm P}_1$”, J. Math. Sci., 73 (1994), 460–467 | DOI | MR

[31] Lukashevich N. A., “The second Painlevé equation”, Differ. Equations, 7 (1971), 853–854

[32] Murata Y., “Rational solutions of the second and the fourth Painlevé equations”, Funkcial. Ekvac., 28 (1985), 1–32 | MR | Zbl

[33] Novokshenov V. Y., “Distributions of poles to Painlevé transcendents via Padé approximations”, Constr. Approx., 39 (2014), 85–99 | DOI | MR | Zbl

[34] Roffelsen P., “Irrationality of the roots of the Yablonskii–Vorob'ev polynomials and relations between them”, SIGMA, 6 (2010), 095, 11 pp., arXiv: 1012.2933 | DOI | MR | Zbl

[35] Roffelsen P., “On the number of real roots of the Yablonskii–Vorob'ev polynomials”, SIGMA, 8 (2012), 099, 9 pp., arXiv: 1208.2337 | DOI | MR | Zbl

[36] Rogers C., Bassom A. P., Schief W. K., “On a Painlevé II model in steady electrolysis: application of a Bäcklund transformation”, J. Math. Anal. Appl., 240 (1999), 367–381 | DOI | MR | Zbl

[37] Shapiro B., Tater M., On spectral asymptotics of quasi-exactly solvable quartic and Yablonskii–Vorob'ev polynomials, arXiv: 1412.3026

[38] Vorob'ev A. P., “On the rational solutions of the second Painlevé equation”, Differ. Equations, 1 (1965), 58–59 | Zbl

[39] Yablonskii A. I., “On rational solutions of the second Painlevé equation”, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk, 1959, no. 3, 30–35