### The Painleve Project

In recent years the Painleve equations, particularly the six Painleve transcendents PI,...,PVI, have emerged as the core of modern special function theory. In the 18th and 19th centuries, the classical special functions such as the Bessel functions, the Airy function, the Legendre functions, the hypergeometric functions, and so on, were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas. In the same way, around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painleve functions, started to appear in applications. The list of problems now known to be described by the Painleve equations is large, varied and expanding rapidly. The list includes, at one end, 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, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar solutions of integrable equations, combinatorial problems such as Ulam's longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems.

Over the years, the properties of the classical special functions -- algebraic, analytical, asymptotic and numerical -- have been organized and tabulated in various handbooks such as the Bateman Project or 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 the properties -- algebraic, analytical, asymptotic and numerical -- of the Painleve functions. This letter is an appeal to interested parties in the scientific community at large for help in developing such a "Painleve Project". What we have in mind, will be described below.

Although the Painleve equations are nonlinear, much is already known about their solutions, particularly their algebraic, analytical and asymptotic properties. This is 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 non-linear steepest-descent method. The numerical analysis of the equations is less developed and presents novel challenges: in particular, in contrast to the classical special functions, where the linearity of the equations greatly simplifies the situation, each problem for the nonlinear Painleve equations arises essentially anew.

As a first step in the Painleve Project, we have established an e-site, maintained at the National Institute of Standards and Technology (NIST). We ask interested readers to send to the site:

(1)   pointers to new work on the theory of the Painleve equations, algebraic, analytical, asymptotic, or numerical;

(2)   pointers to new applications of the Painleve equations;

(3)   suggestions for possible new applications of the Painleve equations;

(4)   requests for specific information about the Painleve 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 email to daniel.lozier@nist.gov.

(2)   To post a message after becoming a subscriber, send email to PainleveProject@nist.gov. The message will be forwarded to every subscriber.

(3)   See Archive for the complete record of posted messages.  This archive is visible to anyone, not just subscribers.

(4)   See Subscribers for the complete list of subscribers. This list is visible to anyone, not just subscribers.

Depending on the response to our appeal, we plan to set up a Wiki page for the Painleve equations, and then ultimately a comprehensive handbook in a style befitting our digital age, along the lines of the hyperlinked version (http://dlmf.nist.gov) 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 Painleve equations.

The Painleve Project was conceived at a workshop held at the International Centre for Mathematical Sciences, Edinburgh, May 10-14, 2010. The Painleve Project has been initiated by F. Bornemann, P. Clarkson, P. Deift, A. Edelman, A. Its, and D. Lozier.

Last change on this page: December 1, 2010