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1: 32.8 Rational Solutions
§32.8 Rational Solutions
Special rational solutions of P III  are … Then P III  has rational solutions iff … These rational solutions have the form …
2: Bibliography Y
  • A. I. Yablonskiĭ (1959) On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).
  • 3: 32.9 Other Elementary Solutions
    These are rational solutions in ζ = z 1 / 3 of the form … These are rational solutions in ζ = z 1 / 2 of the form …
    4: Bibliography K
  • K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.
  • K. Kajiwara and Y. Ohta (1998) Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.
  • A. V. Kitaev, C. K. Law, and J. B. McLeod (1994) Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.
  • 5: Bibliography V
  • A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).
  • 6: Bibliography M
  • T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
  • M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
  • Y. Murata (1985) Rational solutions of the second and the fourth Painlevé equations. Funkcial. Ekvac. 28 (1), pp. 1–32.
  • 7: Bibliography
  • H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.
  • H. Airault (1979) Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.
  • 8: Bibliography C
  • P. A. Clarkson (2005) Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.
  • 9: 31.14 General Fuchsian Equation
    An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …
    10: 28.12 Definitions and Basic Properties
    When ν = s / m is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period 2 m π . …