Mathematical Sciences: Some Questions in Nonlinear Differential Equations

该项目继续对与常微分方程和偏微分方程式有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而获得的解的性质。研究常微分方程式的工作集中在称为哈密顿系统的系统上，该系统用作有限质点系统的运动模型。自20世纪70年代S以来，用变分发展起来的新方法研究哈密顿系统的周期解取得了长足的进展。目前的研究将利用过去的工作来分析其他类型的全局解(周期解是最简单的，但也有其他类型的解)。解的路径，通常被称为轨道，由于失去了有界性，处理起来要困难得多。尽管如此，仍有一些有希望的初步工作可以进一步进行。对于半线性椭圆型偏微分方程，也可以提出类似的问题。早期的研究已经导致了解的存在，这些解的符号不变。虽然现在关于这个主题的文献很多，但结果很少说明解决方案的性质(例如，如何描述解决方案表面的上升和下降？)将就这类性质的问题开展工作。

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.