Folkmar Bornemann, Peter Clarkson, Percy Deift, Alan Edelman, Alexander Its, and Daniel Lozier

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 such as the Bessel functions, the Airy function, the Legendre functions, and the hypergeometric functions were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity, and 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 Painlevé functions—started to appear in applications. 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 =1/2.

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. What is needed now is a comparable organization and tabulation of the properties of the Painlevé functions. This is an appeal to interested parties in the scientific community for help developing the Painlevé Project.

Although the Painlevé 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 that they have a Lax-Pair and 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, particularly in contrast to the classical special functions. Where the linearity of the equations greatly simplifies the situation, each problem for the nonlinear Painlevé equations arises essentially anew.

A site has been established for the Painlevé Project, maintained at the National Institute of Standards and Technology (NIST), where interested readers can send material. Depending on the response to this appeal, a wiki for the Painlevé equations may be set up and a comprehensive online handbook created.

Material being sought includes the following:

• Pointers to new work on the theory of the Painlevé equations—algebraic, analytical, asymptotic, or numerical
• Pointers to new applications of the Painlevé equations
• Suggestions for possible new applications of the Painlevé equations
• Requests for specific information about the Painlevé equations

How to Use the Site

You must be a subscriber to post messages. To become a subscriber, send an email request to Daniel Lozier at daniel.lozier@nist.gov. To post a message, send an email to PainleveProject@nist.gov. The message will be forwarded to every subscriber.

Visit this website for the complete archive of posted messages. This archive is visible to anyone. For a complete list of subscribers, visit the Painlevé Project website. This list is visible to anyone.