New analytic techniques in differential and difference systems

微分和差分系统中的新分析技术

• 批准号: 0100495
• 负责人: Martin Kruskal
• 金额: $ 12万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100495&HistoricalAwards=false
• 关键词: New; analytic; techniques; differential; difference

New methods based on generalized Borel summation and trans-asymptoticexpansions are being developed by a number of analysts and are emergingas a new field of study, "analyzability theory". The methods enable oneto treat solutions of equations at singularities much more general thenusually dealt with and provide a rigorous foundation to previouslyformal calculations. The present proposal aims at developing andapplying these new techniques. The PIs will continue their research inthe context of ODEs, difference equations and PDEs. Applicationsaddressed include the classification of nonlinear ODEs at irregularsingularities, determining their integrability properties as well astheir reducibility to simpler systems, study of adiabatic invariance,and Painleve' analysis for difference equations. These techniques willbe used to continue rigorous research of the time dependent Schroedingerproblem under periodic perturbations of arbitrary strength and also ofresonance theory.The project is a piece in an unfolding enterprise to bridge the gap fromformal calculations to rigorous proofs, and in the advance of whatappears to be emerging as a general theory of singular behavior.Countless applications exist in mathematics as well as in the study ofphysical phenomena.

许多分析师正在开发基于广义 Borel 求和和反渐近展开的新方法，并正在成为一个新的研究领域：“可分析性理论”。这些方法使人们能够比通常处理的更普遍地处理奇点处的方程解，并为以前的正式计算提供严格的基础。 本提案旨在开发和应用这些新技术。 PI 将继续在 ODE、差分方程和 PDE 方面进行研究。涉及的应用包括不规则奇点处的非线性常微分方程的分类、确定其可积性质及其简化为更简单系统的能力、绝热不变性的研究以及差分方程的 Painleve 分析。这些技术将用于继续对任意强度和共振理论的周期性扰动下的时间相关薛定谔问题进行严格研究。该项目是一个正在展开的事业中的一部分，旨在弥合从形式计算到严格证明的差距，并推动似乎正在出现的奇异行为的一般理论的发展。数学和物理研究中存在无数的应用。 现象。