Mathematical Sciences: Some Questions in Nonlinear Differential Equations

数学科学：非线性微分方程的一些问题

• 批准号: 9123265
• 负责人: Paul Rabinowitz
• 金额: $ 14.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-05-15 至 1996-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9123265&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Questions; Nonlinear

This project continues mathematical research on questions related to ordinary and partial differential equations. The research considers properties of solutions obtained by combining global topological techniques with analytic tools. Work on ordinary differential equations focuses on systems, called Hamiltonian systems, which serve as models for the motion of finite systems of particles. Since the 1970's there has been considerable progress in the study of periodic solutions of Hamiltonian systems through newly developed techniques growing out of the calculus of variations. Current research will use past work in analyzing other types of global solutions (the periodic ones are the simplest, but there are others). The solution paths, often called orbits, are much more difficult to treat because of the loss of boundedness. Still, there is some promising preliminary work which can be pursued further. There are analogous questions which can be raised for semilinear elliptic partial differential equations. Earlier research has led to the existence of solutions which do not change sign. Although there is considerable literature on the subject now, the results say little about the nature of the solutions (e.g. how can one describe the rise and fall of solution surfaces?). Work will be done on questions of this nature.

该项目继续对与常微分方程和偏微分方程相关的问题进行数学研究。 该研究考虑了通过将全局拓扑技术与分析工具相结合获得的解决方案的属性。常微分方程的研究重点是称为哈密顿系统的系统，它充当有限粒子系统的运动模型。 自 20 世纪 70 年代以来，通过变分法中新开发的技术，哈密顿系统周期解的研究取得了相当大的进展。 当前的研究将利用过去的工作来分析其他类型的全局解决方案（周期性解决方案是最简单的，但还有其他解决方案）。 由于失去了有界性，解路径（通常称为轨道）更难处理。 尽管如此，仍有一些有希望的初步工作可以进一步开展。 对于半线性椭圆偏微分方程也可以提出类似的问题。 早期的研究已经导致存在不改变符号的解决方案。 尽管现在有大量关于这一主题的文献，但结果很少提及解的性质（例如，如何描述解表面的上升和下降？）。 将针对这种性质的问题开展工作。