Mathematical Sciences: Some Questions in Nonlinear Differential Equations

这个项目继续数学研究的问题 与常微分方程和偏微分方程有关。 的 研究认为，通过结合 全局拓扑技术与分析工具。工作 常微分方程的重点是系统，称为 哈密顿系统，作为模型的运动， 粒子的有限系统 自20世纪70年代以来， 周期解的研究取得了很大进展 通过新开发的技术， 从变分法中得到的。 目前的研究将使用 过去在分析其他类型的全球解决方案方面的工作（ 周期性的是最简单的，但还有其他的）。 的 解路径，通常称为轨道， 因为失去了界限而治疗。 尽管如此， 有希望的初步工作，可以进一步开展。 有类似的问题，可以提出， 半线性椭圆型偏微分方程 早些时候 研究已经导致了解决方案的存在， 变号。 虽然有大量的文献， 现在的主题，结果说关于的性质很少 解决方案（例如，如何描述 溶液表面？）。 关于这方面的问题， 自然

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.