Painlevé Project Seeks Contributors

November 16, 2010

In recent years the Painlevé equations, particularly the six Painlevé transcendents PI, . . . , PVI, have emerged as the core of modern special function theory. Much as the classic special functions of the 18th and 19th centuries---such as the Bessel functions, the Airy function, the Legendre functions, the hypergeometric functions--were recognized and developed in response to problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas, the new Painlevé functions started to appear in applications around the middle of the 20th century, as science and engineering continued to expand in new directions. The list of problems now known to be described by the Painlevé equations is large, varied, and rapidly expanding. At one end is the scattering of neutrons off heavy nuclei and at the other, the statistics of the zeros of the Riemann zeta function on the critical line Re (z) = 1/2. And in between, amongst many others, are random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar solutions of integrable equations, such combinatorial problems as Ulam's longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime in which the number of variables and the number of samples are comparable and large, and random growth problems.

Over the years, the properties of the classic special functions--algebraic, analytical, asymptotic, and numerical--have been organized and tabulated in various handbooks, such as the Bateman Project and the National Bureau of Standards Handbook of Mathematical Functions, edited by Abramowitz and Stegun. What is needed now is a comparable organization and tabulation of those properties for the Painlevé functions. We invite interested parties in the scientific community at large to help in the development of such a "Painlevé project."

Although the Painlevé equations are non-linear, much is already known about their solutions (known collectively as "Painlevé functions"), particularly their algebraic, analytical, and asymptotic properties. This is so because the equations are integrable, in the sense that they have a Lax pair and also a Riemann–Hilbert representation from which the asymptotic behavior of the solutions can be inferred using the nonlinear steepest-descent method. The numerical analysis of the equations is less developed and presents novel challenges: In particular, in contrast to the classic special functions, where the linearity of the equations greatly simplifies the situation, each problem for the nonlinear Painlevé equations arises essentially anew.

As a first step in the Painlevé Project, we have established an e-site, maintained at the National Institute of Standards and Technology, to which interested readers are invited to send: (1) pointers to new work on the theory of the Painlevé equations--algebraic, analytical, asymptotic, or numerical; (2) pointers to new applications of the Painlevé equations; (3) suggestions for possible new applications of the Painlevé equations; and (4) requests for specific information about the Painlevé equations.

The e-site will work as follows: (1) You must be a "subscriber" to post messages to the e-site. To become a subscriber, send e-mail to (2) To post a message after becoming a subscriber, send e-mail to The message will be forwarded to every subscriber. (3) The complete archive of posted messages can be found at Access to this archive is not limited to subscribers. (4) For the complete list of subscribers, see Again, the list can be viewed by anyone.

Depending on the response to this appeal, we plan to set up a wiki for the Painlevé equations and, ultimately, a comprehensive handbook along the lines of the hyperlinked version ( of the new NIST Handbook of Mathematical Functions, edited by Olver, Lozier, Boisvert, and Clark and published by Cambridge University Press. Incidentally, this work contains, for the first time, a chapter on the Painlevé equations.---F. Bornemann, P. Clarkson, P. Deift, A. Edelman, A. Its, and D. Lozier.

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