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Painleve equations: analytical properties and numerical computation

Painleve 方程：分析特性和数值计算

基本信息

批准号：
EP/P026532/1
负责人：
Alfredo Deaño
金额：
$ 12.17万
依托单位：
University of Kent
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP026532%2F1
关键词：
Painleve equations analytical properties numerical

项目摘要

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.

该项目研究Painlevé II和IV微分方程解的解析和数值分析的新工具。这些微分方程的某些解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发挥着关键作用，其次是因为它们的数值计算对数值输入数据特别敏感。作为例子，tronquée解和特殊函数解在这方面尤为重要。从广义上讲，该项目属于数学物理特殊函数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评估数学函数，从初等函数（指数和对数，三角和双曲）到所谓的经典特殊函数（包括Gamma和误差函数，Airy，Bessel，抛物柱面函数和超几何函数家族的一般成员）。许多这样的方法已经在数值和符号软件（Matlab，Maple，Mathematica）的标准包中实现，并且是Fortran，C或Python等语言的核心库的一部分。Painlevé方程是二阶非线性常微分方程分类的一般问题的结果，该方程具有所有解都没有可移动（取决于初始条件）分支点的性质。由Painlevé和Gambier发起，这项工作最终列出了六个这样的方程（直到变量的变换和变化），称为Painlevé方程。它们的解通常被称为Painlevé超越，或非线性特殊函数，因为它们所产生的微分方程的非线性特征。在过去的几十年中，他们发现了越来越丰富的各种应用，从随机矩阵理论到组合数学，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，特别是在复平面上，直到几年前，计算它们的唯一通用方法是使用常微分方程的数值方法，无论是以初始值的形式还是边值问题。这种方法被Fornberg和Weideman，Fornberg和Reeger和Bornemann利用。直到最近才用于数值计算的一个重要信息是，Painlevé trascendents可以用某些黎曼-希尔伯特问题（RHP）的解来描述，这些问题是复平面上的边值问题。这个强大的公式已经打开了一个新的世界的可能性，它现在是一个重要的工具，在理论，渐近和数值分析的Painlevé方程。这个项目将建立在这些想法，扩大他们和调查他们的适用性，以获得分析和数值信息的解决方案Painlevé II和IV的兴趣。这项任务意味着现有的理论和广泛的测试，这些数值算法的实质性修订和扩展。