Painleve equations: analytical properties and numerical computation

This project studies new tools for the analytical and numerical analysis of solutions of the Painlevé II and IV differential equations. Certain families of solutions of these differential equations are especially relevant in our work, firstly because they play a key role in areas like random matrix theory, orthogonal polynomials and integrable systems, and secondly because their numerical computation is especially delicate and sensitive to numerical input data. As examples, tronquée solutions and special function solutions are particularly important in this context.

In a broad sense, the project belongs to the general area of numerical calculation of special functions of mathematical physics, which has been a very active field of research for decades in numerical analysis and applied mathematics. Since the advent of modern computers, many algorithms have been devised to evaluate mathematical functions in a reliable way, ranging from the elementary ones (exponential and logarithmic, trigonometric and hyperbolic) to the so-called classical special functions (including the Gamma and error functions, Airy, Bessel, parabolic cylinder functions and in general members of the family of hypergeometric functions). Many such methods are already implemented in the standard packages of numerical and symbolic software (Matlab, Maple, Mathematica) and are part of core libraries in languages like Fortran, C or Python.

The Painlevé equations are the result of the general problem of classification of second order nonlinear ordinary differential equations that have the property that all the solutions are free of movable (depending on initial conditions) branch points. Initiated by Painlevé and Gambier, this work led to a final list of six such equations (up to transformations and changes of variables) that are called the Painlevé equations. Their solutions are often referred to as Painlevé transcendents, or nonlinear special functions, because of the nonlinear character of the differential equations that they arise from. During the last decades, they have found an increasingly rich variety of applications, from random matrix theory to combinatorics, number theory and partial differential equations. Because of their nonlinear origin, they also pose new analytical and numerical challenges, particularly in the complex plane, and up to a few years ago the only general approach to compute them was to use numerical methods for ordinary differential equations, either in the form of initial value or boundary value problems. This approach was exploited by Fornberg and Weideman, Fornberg and Reeger and Bornemann. An essential piece of information that was not used for numerical work until recently is the fact that Painlevé trascendents can be described in terms of the solution of certain Riemann-Hilbert problems (RHP), which are boundary value problems in the complex plane. This powerful formulation has opened a new world of possibilities and it is now an essential tool in the theoretical, asymptotic and numerical analysis of the Painlevé equations.

This project will build on these ideas, expanding them and investigating their applicability to obtain analytical and numerical information about the solutions of Painlevé II and IV that are of interest. This task implies a substantial revision and extension of the existing theory and also extensive testing of those numerical algorithms.

It was clearly stressed in the 2012 Report "Measuring the Economic Benefits of Mathematical Research in the UK", that research in the mathematical sciences is a vital aspect of modern society, and that in particular contributes crucially to the strength of the UK economy. The applications of mathematical research (in a broad sense) are often not immediate, but they range from enhanced engineering devices to models for economic, biological and environmental systems or cryptography.

The proposed project will deliver design and analysis of accurate and reliable methods for the evaluation of special functions (of Painlevé type). This area has been extremely active for decades, often driven by the need to have reliable software routines to solve problems in partial differential equations, optimization and mathematical models. Not only mathematicians, but also physicists and computer scientists have contributed to the design, implementation and verification of toolboxes and computer packages in different programming languages, both in the academic and in the non-academic environment. The fact that the classical Handbook of Mathematical Functions by Abramowitz and Stegun has been recently revised and expanded (including numerical methods) to produce the Digital Library of Mathematical Functions (DLMF), published by the National Institute of Standards and Technology (NIST) in the United States, which is not a purely academic institution, is a proof of the importance of this area. This publication was made freely available on the web at http://dlmf.nist.gov, and is regularly updated and used by millions of people in and outside the academic world. We stress that the section in the DLMF on numerical methods for the Painlevé equations is at present very limited, and therefore any significant advancement in this direction will have a major impact in the whole community of users.

Additionally, NAG (Numerical Algorithms Group) has shown interest in the computation of special functions for quite a long time, and has incorporated dedicated algorithms into their software routines. They are aware of the increasing importance of the Painlevé equations and actively interested in their numerical implementation. An important part of the impact that will result from this project will consist of exploiting this collaboration.

Impact includes direct training of personnel as well, namely a dedicated Post-Doctoral Research Assistant (PDRA) at Kent, who will play an essential role in the development of the project. The combination of analytical and numerical skills that he/she will develop will greatly enhance his/her future employability, whether in academia or in industry.