中科院数学与系统科学研究院
数学研究所
动力系统研究中心
学术报告会
报告人：Prof. Rod Halburd（University College London，UK）
题 目：Painlevé-like properties for differential, discrete and delay-differential equations
时 间：2016.09.22（星期三）, 15:00-16:00
地 点：数学院南楼N913室
Abstract：
It has long been known that the singularity structure of solutions of an ordinary differential equation (ODE) in the complex domain gives a strong indicator of its integrability. Equations with the Painlevé property (that all solutions are single-valued about all movable singularities) are particularly important in this respect. I will discuss a number of generalisations of this property.
Methods for classifying particular solutions of ODEs such that all movable singularities are poles will be discussed. This involves a more delicate global analysis of particular solutions than is the case in Painlev\'e analysis. In standard Painlevé analysis it is sufficient to find movable branching in any solution in order to discard an equation.
If the fixed singularities are also no worse than poles then Nevanlinna theory provides the necessary global tool. In this talk, new methods will be applied so as to allow for branching at fixed singularities. Methods for classifying solutions with movable branch points such that globally the number of sheets is bounded will also be described.
Finally, the singularity structure of solutions of various discrete and delay-differential equations will be used to detect apparently integrable cases. The tools used include Nevanlinna theory in the complex plane and the p-adic absolute value of solutions over the rational numbers.
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