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）几何和（c）偏微分方程中的一些问题。对于（a），基于一些最近的研究，我们用变分方法构造了2-torus和n-torus上Hamilton系统的各种同宿和异宿轨道。当一个额外的对称性存在时，环面。当寻找与n-环面上的一对周期测地线异宿的最小测地线时，对（B）正在追求一些类似的问题。对（a）感兴趣的另一个问题是，是具有奇异位势的Hamilton系统的同宿解的存在性，特别是当位形空间的维数大于2时。最后对于（c），最近克雷格和韦恩以及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进展。我们感兴趣的是变分工具是否可以用于在这种设置中获得大幅度的解决方案。 动力系统和偏微分方程用于模拟科学、工程、经济和许多其他领域中出现的连续现象。令人感兴趣的是对所有时间变量值存在的此类方程的解。除了平衡点，即与时间无关的解，这种类型的最简单的解是周期解。周期解在一个时间段后返回到它们的原始状态。下一个最简单的类时间上的全局解是所谓的异宿解，在最简单的情况下，当时间向前或向后无穷大时，该异宿解将接近一对不同的平衡或时间周期解。(When我们的研究涉及到动力系统的异宿解和同宿解的求解方法以及某些特殊类型的偏微分方程的周期解的求解。