Some Questions in Nonlinear Differential Equations

DMS-9732529PI：Rabinowitz我们计划研究(A)动力系统、(B)几何和(C)偏微分方程组中的一些问题。对于(A)，在最近的一些研究的基础上，我们正在使用变分方法来构造2-环面上的哈密顿系统的各种同宿和异宿轨道，以及当存在附加对称性时在n-环面上的哈密顿系统的各种同宿和异宿轨道。对于(B)当n-环面上的一对周期测地线的极小测地线异宿时，我们正在探索一些类似的问题。(A)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配置空间的维度大于2时。最后，对于(C)，最近Craig和Wayne以及Bourain在非线性波动方程的小振幅周期解的存在性方面取得了一些进展，我们感兴趣的是在这种情况下是否可以使用变分工具来获得大振幅解。动力系统和偏微分方程被用来模拟科学、工程、经济和许多其他领域中出现的连续现象。相当有趣的是这种方程的解对于时间变量的所有值都存在。除了平衡，即与时间无关的解，这种类型的最简单的解是周期解。周期解在一个时间周期后返回到它们的原始状态。在最简单的情况下，当时间向前或向后到无穷大时，全局在时间上的下一类最简单的解是所谓的异宿解，它将接近一对不同的平衡或时间周期解。(当极限状态相同时，我们的研究包括发展寻找动力系统的异宿解和同宿解的方法，以及寻找某些特殊类型的偏微分方程周期解的方法。

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.