Painleve Equations

摘要本课题的目的是研究painlevel微分方程的一些基本问题。这些方程是二阶非线性微分方程，其解没有所谓的可动奇点，不能用初等函数或特殊函数表示。它们的重要性源于Painleve性质与可积性理论的联系，以及Painleve超越解的众多应用。在这个项目中，主要研究者将研究painleve相关分析的以下几个方面：单值亚纯解的生长阶；painlevel方程解的值分布与分支和painlev型方程，这些方程允许可移动的分支点，但只允许其多重性由预先指定的常数限定。在这个建议下进行的工作将导致对这类微分方程的更好的理解和具体的结果。疼痛级方程在纯数学和应用数学以及数学在其他科学和工程中的应用中具有重要意义。在数学中，疼痛级方程被应用于微分几何、随机矩阵模型和可积性。以下是目前在数学之外的活动领域的例子，在这些领域中，疼痛级方程被发现是有用的，并且以一种自然的方式出现：物理学中的Ising模型、弹性学中的统计力学、反铁磁体模型中的相关函数、量子场论和拓扑场论、广义相对论和宇宙学、物理学中的超对称规范理论、浅水中的共振振荡、粘性流体中的Hele-Shaw问题、等离子体物理学、超导、非线性光学和光纤、聚合物、聚电解质和胶体。这个列表本身就清楚地表明了数学在科学和工程领域的强大影响，以及数学中的理论理解和精确解决问题可以为其他科学的建模和理论构建做出贡献的价值。这个项目将为这个广泛适用的领域提供额外的知识和方法。

ABSTRACTThe purpose of this project is to study a number of fundamental problems for the Painleve differential equations. These equations are second order nonlinear differential equations whose solutions do not have so-called movable singularities and cannot be expressed in terms of elementary or special functions. Their importance arises from the connections of the Painleve property to integrability theory, as well as from the numerous applications of the solutions, the Painleve transcendents. In this project, the principal investigator will study the following aspects of Painleve-related analysis: order of growth of single-valued meromorphic solutions; value distribution and branching of solutions to Painleve equations; andPainleve-type equations which admit movable branch points but only those whose multiplicity is bounded by a preassigned constant. Work performed under this proposal will lead to a greater understanding of and concrete results for this class of differential equations.Painleve equations are of great importance in pure and applied mathematics as well as inthe applications of mathematics to other sciences and to engineering. Within mathematics,Painleve equations are being applied in differential geometry, random matrix models, and integrability. The following are examples of areas of current activity outside mathematics in which Painleve equations have been found useful and have arisen in a natural way: the Ising model in physics, statistical mechanics in elasticity, correlation functions in an antiferromagnet model, quantum field theory and topological field theory,general relativity and cosmology, supersymmetry gauge theories in physics,resonant oscillations in shallow water, Hele-Shaw problems in viscous fluids,plasma physics, superconductivity, nonlinear optics and fiber optics,polymers, polyelectrolytes, and colloids. This list alone provides a clear indication of the empowering impact of mathematics in science and engineering, and of the value that theoretical understanding and precise problem solving in mathematics can contribute to modeling and theory building in other sciences. This project will provide additional knowledge and methods to this area of wide applicability.