Some Questions in Nonlinear Differential Equations

非线性微分方程的一些问题

• 批准号: 9732529
• 负责人: Paul Rabinowitz
• 金额: $ 8.7万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-15 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732529&HistoricalAwards=false
• 关键词: Some; Questions; Nonlinear; Differential; Equations

DMS-9732529PI: Rabinowitz We plan to study some problems in (a) dynamical systems,(b) geometry,and(c) partial differential equations.For (a),building on some recent research,we are using variational methods to construct a variety of homoclinic and heteroclinic orbits for Hamiltonian systems on the 2-torus and also on the n-torus when an additional symmetry is present.Some analogous questions are being pursued for (b) when minimal geodesics heteroclinic to a pair of periodic geodesics on the n-torus are being sought.Another problem of interest for (a) is the existence of homoclinic solutions for Hamiltonian systems which possess singular potentials, especially when the dimension of the configuration space is greaterthan two.Lastly for (c),there has been some recent progress by Craig and Wayne and by Bourgain on the existence of small amplitude periodic solutions for nonlinear wave equations.We are interested in whether variational tools can be employed to obtain large amplitude solutions in such settings. Dynamical systems and partial differential equations are used to model continuous phenonena arising in science,engineering,economics and many other fields.Of considerable interest are solutions of such equations that exist for all values of the time variable.Aside from equilibria,i.e. time independent solutions,the simplest solutions of this type are periodic ones.Periodic solutions return to their original state after one time period.The next simplest class of global in time solutions are so-called heteroclinic solutions which in the simplest cases will approach a pair of different equilibria or time periodic solutions as time goes forward or backward to infinity.(When the limit states are the same,the solutions are called homoclinics).Our research involves developing methods to find heteroclinic and homoclinic solutions for dynamical systems and finding periodic solutions for some special classes of partial differential equations.

DMS-9732529PI：Rabinowitz 我们计划研究 (a) 动力系统、(b) 几何学和 (c) 偏微分方程中的一些问题。对于 (a)，基于最近的一些研究，我们正在使用变分方法构建当存在附加对称性时，哈密顿系统在 2-环面和 n-环面上的各种同宿和异宿轨道。当最小时，正在针对 (b) 寻求一些类似的问题正在寻找与 n 环面上的一对周期测地线异宿的测地线。 (a) 的另一个感兴趣的问题是具有奇异势的哈密顿系统的同宿解的存在性，特别是当构型空间的维数大于 2 时最后，对于 (c)，Craig 和 Wayne 以及 Bourgain 在非线性波动方程小振幅周期解的存在性方面取得了一些最新进展。我们感兴趣的是是否可以采用变分工具来获得大振幅周期解。此类设置中的幅度解决方案。 动力系统和偏微分方程用于模拟科学、工程、经济学和许多其他领域中出现的连续现象。人们相当感兴趣的是此类方程的解，这些方程对于时间变量的所有值都存在。除了平衡态之外，即与时间无关的解，这种类型最简单的解是周期性解。周期性解在一段时间后返回到其原始状态。下一个最简单的全局时间解是所谓的异宿解，在最简单的情况下将接近一对随着时间向前或向后无穷远，不同的平衡或时间周期解。（当极限状态相同时，这些解称为同宿）。我们的研究涉及开发寻找动力系统异宿和同宿解的方法以及寻找周期解为了一些特殊类别的偏微分方程。