Non-linear equations and Schroedinger operators

该项目的目标是数学物理和偏微分方程理论中的基本问题。它涉及对某些非线性微分方程（例如，色散管理孤子的衰减特性、矩阵薛定谔算子的光谱特性以及极端密度下库仑物质的特性）。在这项研究中使用和开发的方法是从微分方程，分析，谐波分析，变分法，泛函分析和谱理论的分析技术的组合。这个研究项目的主要部分是跨学科的。它连接了纯数学和应用科学，如工程和物理。例如，色散管理孤子现在广泛用于高速互联网电缆。尽管在商业上取得了成功，但人们对这些孤子的衰变知之甚少。工程师关心衰减，因为它有效地决定了光纤通信设备的带宽（因此速度）。这些研究将有助于更好地理解这些孤子。该项目中开发的新数学方法不仅将反馈到数学家可用的分析工具箱中，而且还将为工程师经常进行的临时正式计算奠定坚实的基础。这可能会加深对色散方程数学基础的了解，在此过程中为工程师提供了探索这些系统性质的新工具，并增加了改进现有光纤系统的潜力。该项目将扩大和加强与美国，欧洲和韩国研究人员的现有合作。部分赠款将用于支持和培训一名研究生。

This project targets basic questions in mathematical physics and the theory of partial differential equations. It involves a rigorous study of properties of solutions of certain nonlinear differential equations (e.g., the decay properties of dispersion managed solitons, spectral properties of matrix Schrödinger operators, and properties of Coulomb matter under extreme densities) that correspond to significant physical phenomena. The methods used and developed in this research are a combination of analytic techniques from differential equations, analysis, harmonic analysis, the calculus of variations, functional analysis, and spectral theory. The major part of this research project is interdisciplinary. It bridges pure mathematics and applied sciences such as engineering and physics. For example, dispersion management solitons are by now widely used in high-speed internet cables. Despite this commercial success only very little is known rigorously about the decay of these solitons. Engineers care about the decay, because it effectively determines the bandwidth (hence the speed) of optical fiber communication devices. The research will lead to a better rigorous understanding of these solitons. Not only will the new mathematical methods developed in the project feed back into the analytic toolbox available for mathematicians, but it will also put the often ad-hoc formal calculations of engineers on a firm basis. This could lead to deepened insights into the mathematical foundations of dispersive equations, in the process providing engineers with both new tools for exploring the nature of these systems and increased potential for improving existing fiber optics systems. The project will extend and strengthen existing collaborations with researchers in the USA, Europe, and Korea. Part of the grant will be used for supporting and training a graduate student.