Painleve equations: analytical properties and numerical computation

本项目研究用于分析和数值分析PainlevéII和IV微分方程解的新工具。这些微分方程解的某些族在我们的工作中特别重要，首先是因为它们在随机矩阵理论、正交多项式和可积系统等领域中起着关键作用，其次是因为它们的数值计算特别精细和对数值输入数据敏感。广义上讲，本课题属于数学物理中特殊函数的数值计算的一般领域，几十年来一直是数值分析和应用数学中非常活跃的研究领域。自从现代计算机出现以来，已经设计了许多算法来以可靠的方式计算数学函数，从初等函数(指数和对数、三角和双曲线)到所谓的经典特殊函数(包括伽马函数和误差函数、艾里函数、贝塞尔函数、抛物线柱面函数和一般超几何函数族的成员)。许多这样的方法已经在数值和符号软件的标准包中实现(MatLab、Maple、数学a)，并且是Fortran、C或Python语言的核心库的一部分。Painlevé方程是一般的二阶非线性常微分方程组分类问题的结果，它具有所有解都不存在可移动(取决于初始条件)分支点的性质。这项工作由Painlevé和Gbier发起，最终形成了六个这样的方程(直到变量的变换和变化)，被称为Painlevé方程。它们的解通常被称为PainlevéTranscendents，或非线性特殊函数，因为它们产生的微分方程具有非线性特性。在过去的几十年里，它们发现了越来越丰富的应用，从随机矩阵理论到组合学，数论和偏微分方程。由于它们的非线性起源，它们也带来了新的分析和数值挑战，特别是在复杂的平面上，直到几年前，计算它们的唯一一般方法是使用常微分方程组的数值方法，无论是初值问题还是边值问题。这种方法被Fornberg和Weideman，Fornberg和Reeger以及Bornemann所利用。直到最近才被用于数值工作的一条基本信息是，Painlevétrasendent可以用某些Riemann-Hilbert问题(RHP)的解来描述，RHP是复平面上的边值问题。这个强大的公式打开了一个新的可能性世界，现在它是Painlevé方程的理论、渐近和数值分析的基本工具。本项目将基于这些想法，扩展它们并调查它们的适用性，以获得有关PainlevéII和IV解的分析和数值信息。这项任务意味着对现有理论的实质性修改和扩展，以及对这些数值算法的广泛测试。

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.