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Painlevé property

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1: 32.16 Physical Applications
§32.16 Physical Applications
2: Mark J. Ablowitz
Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …
3: 32.2 Differential Equations
An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … There are fifty equations with the Painlevé property. …
4: Bibliography G
  • B. Grammaticos, A. Ramani, and V. Papageorgiou (1991) Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.
  • 5: Bibliography B
  • T. Bountis, H. Segur, and F. Vivaldi (1982) Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A (3) 25 (3), pp. 1257–1264.
  • 6: Bibliography F
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • 7: Bibliography S
  • B. I. Suleĭmanov (1987) The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).
  • 8: Bibliography C
  • J. M. Carnicer, E. Mainar, and J. M. Peña (2020) Stability properties of disk polynomials. Numer. Algorithms.
  • P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
  • P. A. Clarkson and E. L. Mansfield (2003) The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.
  • P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.
  • C. M. Cosgrove (2006) Chazy’s second-degree Painlevé equations. J. Phys. A 39 (39), pp. 11955–11971.
  • 9: 32.11 Asymptotic Approximations for Real Variables
    §32.11(i) First Painlevé Equation
    §32.11(ii) Second Painlevé Equation
    §32.11(iii) Modified Second Painlevé Equation
    §32.11(iv) Third Painlevé Equation
    §32.11(v) Fourth Painlevé Equation
    10: Bibliography H
  • D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
  • S. P. Hastings and J. B. McLeod (1980) A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.
  • N. J. Hitchin (1995) Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.
  • P. Holmes and D. Spence (1984) On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.
  • J. Humblet (1984) Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.