Skip to main content
Article thumbnail
Location of Repository

The fourth Painleve equation

By Peter Clarkson


The six Painleve equations (PI–PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear second-order ordinary differential equations. During the past 30 years there has been considerable interest in the Painleve equations primarily due to the fact that they arise as symmetry reductions of the soliton equations which are solvable by inverse scattering. Although first discovered from pure mathematical considerations, the Painleve equations have arisen in a variety of important physical applications.\ud \ud The Painleve equations may be thought of as nonlinear analogues of the classical special functions. They have a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. The Painleve equations also admit symmetries under affine Weyl groups which are related to the associated B¨acklund transformations. These can be used to generate hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further solutions of the Painleve equations have some interesting asymptotics which are use in applications. In this paper I discuss some of the remarkable properties which the Painleve equations possess using the fourth Painleve equation (PIV) as an illustrative example

Topics: QA372, QA351
Publisher: World Scientific
Year: 2008
OAI identifier:

Suggested articles


  1. (2005). A 38, doi
  2. (1992). A 437,
  3. (1997). A Compendium of Nonlinear Ordinary Differential Equations doi
  4. (2002). de Gruyter, doi
  5. (1991). From Gauss to Painlev´ e: a Modern Theory of Special Functions, doi
  6. (1991). From Gauss to Painleve´: a Modern Theory of Special Functions, doi
  7. (1992). in Painleve´ Transcendents, their Asymptotics and Physical Applications, doi
  8. (1980). Nuovo Cim. doi
  9. (1956). Ordinary Differential Equations doi
  10. (2004). Painlev´ e Equations through Symmetry,
  11. (2004). Painleve´ Equations through Symmetry, doi
  12. (2006). Painleve´ Transcendents: The Riemann-Hilbert approach, doi
  13. (1988). Physica D30, doi
  14. (1999). Special Functions (C.U.P., doi
  15. (1996). Special Functions. An Introduction to the Classical Functions of doi
  16. (1986). The Isomonodromic Deformation Method in the Theory of Painleve´ equations, doi
  17. (1959). Vesti Akad.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.