Mathematical Sciences: Some Questions in Nonlinear Differential Equations

这个项目继续对与常微分方程和偏微分方程有关的问题进行数学研究。研究了将全局拓扑技术与解析工具相结合得到的解的性质。常微分方程的研究主要集中在称为哈密顿系统的系统上，它作为有限粒子系统运动的模型。自20世纪70年代以来，通过从变分法中发展出来的新技术，哈密顿系统周期解的研究取得了相当大的进展。目前的研究将利用过去的工作来分析其他类型的全局解（周期性解是最简单的，但还有其他的）。解路径，通常称为轨道，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开展。对于半线性椭圆型偏微分方程，也可以提出类似的问题。早期的研究已经得出不改变符号的解的存在。虽然现在有相当多的关于这个问题的文献，但结果很少提到溶液的性质（例如，如何描述溶液表面的上升和下降？）。将对这类性质的问题进行研究。

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.