Non-linear equations and Schroedinger operators

这个项目的目标是数学物理和偏微分方程式理论的基本问题。它涉及到对某些非线性微分方程解的性质的严格研究(例如，色散管理孤子的衰减性质，矩阵薛定谔算子的光谱性质，以及极端密度下库仑物质的性质)，这些性质对应于重要的物理现象。在这项研究中使用和发展的方法是来自微分方程组、分析、调和分析、变分、泛函分析和谱理论的分析技术的组合。这项研究项目的主要部分是跨学科的。它架起了纯数学和工程学、物理学等应用科学的桥梁。例如，色散管理孤子现在被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却知之甚少。工程师们关心衰减，因为它有效地决定了光纤通信设备的带宽(因此速度)。这项研究将使人们对这些孤子有更严格的理解。该项目中开发的新数学方法不仅将反馈给数学家可用的分析工具箱，而且还将使工程师经常临时进行的正式计算建立在坚实的基础上。这可能会加深对色散方程数学基础的洞察，在此过程中为工程师提供探索这些系统本质的新工具，并增加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。助学金的一部分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.