Painleve Equations

摘要：6个painlevel微分方程是无可动奇点二阶微分方程的原型。在这个项目中，首席研究员将研究以下painlevel方程和painlevel相关分析的基本问题。首先，对剩余开放情况下painlevel方程解的painlevel性质进行严格证明。其次，求单值亚纯解的通常增长阶的尖锐上界和下界，以及求多值解的其他自然增长测度的尖锐上界和下界，并确定出现小于极大增长的情况。第三，特别由微分几何驱动的应用导致了painlee型微分方程的分类问题，这些微分方程允许具有可移动的分支点，但受其多重性的限制。首席研究员将朝着这样的分类工作，并确定在应用中出现的某些特定方程是否具有这种性质。在过去的十年里，经过几十年的沉寂期，由于发现了大量的跨学科联系和应用，出现了大量关于Painleve方程及其推广的文献。在理论方面，Painleve性质与非线性常微分方程和偏微分方程的可积性概念有关。在纯数学中的其他应用包括微分几何和随机矩阵模型。在跨学科方面，Painleveequations的应用包括：物理学中的Ising和反铁磁体模型、弹性学中的统计力学、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、超导、非线性光学和光纤、聚合物和聚电解质。这清楚地展示了数学在科学和工程领域的推动力量，以及数学中的理论理解和精确的问题解决方法可以为其他领域的建模和理论构建提供价值。在此建议下进行的工作将导致对这类微分方程的更深入的理解和具体的结果，这些方程在其他数学领域以及其他科学和工程领域的许多应用中都得到了应用。

PI: Aimo Hinkkanen, University of Illinois at Urbana-ChampaignDMS-0200752Abstract:The six Painleve differential equations are prototypes of second order differential equations which do not have movable singularities. In this project, the principal investigator will study the followingfundamental problems for the Painleve equations and Painleve-related analysis. Firstly, the task of obtaining rigorous proofs for the Painleve property of the solutions to Painleve equations in the remaining open cases. Secondly, the question of finding sharp upper and lower bounds for the usual order of growth of single-valued meromorphic solutions, and for other natural measures of growth for multi-valued solutions, as well as the determination of the cases where less than maximal growth occurs. Thirdly, applications motivated particularly by differential geometry lead to the question of classifying Painleve-type differential equations which are allowed tohave movable branch points subject to restrictions on their multiplicity. The principal investigator will work towards such a classification and to determine whether certain particular equations arising in applications have this property.Over the last ten years, after a quiet period of many decades, an enormous amount of literature has appeared on the Painleve equations and their generalizations, due to a great number of interdisciplinary connections and applications that have been found. On the theoretical side, the Painleve property is related to the concept of integrability for non-linear ordinary and partial differential equations. Other applications in pure mathematics include differential geometry and random matrix models. On the interdisciplinary side, the applications of Painleveequations include the following: the Ising and antiferromagnet models in physics,statistical mechanics in elasticity, quantum field theory and topological field theory,general relativity and cosmology, supersymmetric gauge theories in physics,resonant oscillations in shallow water, Hele-Shaw problems in viscous fluids,plasma physics, superconductivity, nonlinear optics and fiber optics,polymers, and polyelectrolytes. This provides a clear manifestation of the enablingpower of mathematics in science and engineering, and of the value that theoretical understanding and precise problem solving in mathematics can add to modeling and theory buildingelsewhere. Work performed under this proposal will lead to a greater understanding of and concrete results for this class of differential equations which is being used in numerousapplications in other areas of mathematics as well as in other sciences and engineering.