In recent years the Painlevé equations, particularly the six Painlevé transcendents PI-PVI, have emerged as the core of modern special-function theory. In the 18th and 19th centuries, the classical special functions, including the Bessel, Legendre, Airy, and hypergeometric functions, were recognized and developed in response to problems in electromagnetism, acoustics, hydrodynamics, elasticity, and many other areas. similarly, around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of equations, the Painlevé equations, and their solutions, the Painlevé functions, started to appear in applications. The equations are second order and nonlinear. 1
The list of problems now known to be described by the Painlevé 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 = . Included in between are 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 random growth problems.
Over the years the properties—algebraic, analytical, asymptotic, and numerical—of the classical special functions have been organized and tabulated in various handbooks such as the Bateman Project or the 1964 National Bureau of Standards Handbook of Mathematical Functions, edited by Milton Abramowitz and Irene Stegun. What is needed now is a comparable organization and tabulation of the same properties of the Painlevé functions. This letter is an appeal to interested parties in the scientific community at large for help in developing such a “Painlevé Project.”
Although the Painlevé equations are nonlinear, much is already known about their solutions, particularly their algebraic, analytical, and asymptotic properties. That is because the equations are integrable in the sense that they have a Lax pair and a Riemann-Hilbert representation. A Riemann-Hilbert representation is a nonlinear analog of the familiar integral representations of the classical special functions. And just as one can apply these integral representations to determine the asymptotic behavior of those functions using the classical method of steepest-descent, so too there exists a nonlinear steepest-descent method which can be used to determine the asymptotic behavior of the Painlevé functions, with equal efficiency and accuracy. The numerical analysis of the equations, however, is less developed and presents novel challenges. In particular, in contrast to the classical special functions, for which the linearity of the equations greatly simplifies the calculations, each problem for the nonlinear Painlevé equations arises essentially anew.
As a first step in the Painlevé Project, we have established a webpage (http://math.nist.gov/~lozier/PainleveProject/), maintained by NIST. We ask interested readers to visit the site, subscribe, and report new work on the theory of the Painlevé equations, whether algebraic, analytical, asymptotic, or numerical. Users can also request specific information about solutions of the equations and draw attention to possible new applications.
