Painleve equations: analytical properties and numerical computation

点击翻译按钮获取中文摘要

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.