Mathematical Sciences: Some Questions in Nonlinear Differential Equations

Rabinowitz将研究各种类型哈密顿系统的基本同宿解的存在性，如奇异系统，似乎有解同宿无穷大的系统，以及具有几乎周期势的系统。一旦确定了基本同斜，最近的变分方法表明，人们也可以找到无限多个所谓的多凸解，它们是基本解的平移的近似和。这是Rabinowitz要探讨的另一个问题。最后，对于偏微分方程，他将研究一些非线性波方程和薛定谔方程，它们几乎具有周期势。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

PI: Rabinowitz DMS-9500570 Rabinowitz will investigate the existence of basic homoclinic solutions of Hamiltonian systems of various types such as singular systems, systems where there seem to be solutions homoclinic to infinity, and systems with almost periodic potentials. Once basic homoclinics have been determined, recent variational methods suggest that one can also find infinitely many so-called multibump solutions which are near sums of translates of the basic solutions. This is another question Rabinowitz will pursue. Finally, for PDE's he will look at some nonlinear wave and Schrodinger equations with almost periodic potentials. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.