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.
