Galois方法在数学物理问题中的应用

项目名称： Galois方法在数学物理问题中的应用

项目编号： No.11301210

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 黎文磊

作者单位： 吉林大学

项目金额： 23万元

中文摘要： 19世纪初，Galois为解决一般多项式方程的根式可解性而建立了著名的Galois理论。随后，人们不断尝试运用类似的方法研究微分方程的积分可解性,并取得了一系列重要结果： Lie建立了微分方程的 Lie群理论，Picard与Vissiot等人建立的关于线性微分方程的微分Galois理论等；上世纪末，Morales与Ramis等人利用Galois方法建立了复解析Hamilton系统的可积性理论，并被广泛地应用于研究各类数学物理问题的可积性及相关问题；最近，我们利用Galois方法对一般非线性系统给出了类似的可积性理论。本项目中，我们将利用Galois方法研究几类数学物理问题(特别是Hamilton系统)的不可积性与复杂行为之间的关系，同时尝试利用Galois方法研究一类无穷维发展方程及Painlevé方程的可积性及相关问题。

中文关键词： 微分Galois理论；Painleve方程；发展方程；复杂性；不变流形

英文摘要： In early 19th century, in order to solve the problem of radical solvability of general polynomial equations, Galois established the famous Galois theory. From that on, people kept trying to study the integrability of differential equations with similar methods as the Galois theory.Following this way, a series of important results have been obtained: Lie established the Lie group theory for differential equations, Picard and Vissiot got the differential Galois theory for integrability of linear differential equations, which is also called the Picard-Vissiot theory; in the end of last century, Morales and Ramis used differential Galois approach to get an effectual integrability theory for complex analytic Hamiltonian system, which is now widely applied to study the integrability of the various mathematical and physical problems; recently, we used the Galois method to get a similar integrability theory for general nonlinear systems. In this project, we will use the Galois method to study the relationship between the non-integraility and complex behavior for several classes of mathematical and physical problems (in particular the Hamiltonian systems), meanwhile, we will also try to take advantage of the Galois method to study the non-integrability of a class of infinite dimensional evolution equations and Pa

英文关键词： Differential Galois theory；Paileve equations；Evolution equations；Complexity；Invariant manifolds