Harmonic Analysis and Nonlinear Dispersive Equations

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

Harmonic Analysis and Nonlinear Dispersive Equations Abstract of Proposed Research Atanas G Stefanov This project centers on the use of methods from Fourier analysis to analyze certain nonlinear partial differential equations arising in mathematical physics. Our primary interest is on dispersive equations with either Schroedinger or wave operators as the leading order terms. In particular the Maxwell-Schroedinger, Maxwell-Klein-Gordon and the Maxwell-Dirac systems will be studied. Only ``rough'' initial data will be assumed. This is more realistic physically and requires more careful use of the underlying conservation laws. We want to obtain results about the well-posedness, regularity and asymptotic behavior of the solutions of these equations.. We expect that new tools from Fourier analysis, spectral theory and gauge theoretic methods will be developed and used to prove these results. The current proposal will produce new mathematical tools to study nonlinear dispersive equations, which are mathematical models for important processes, such as magnetization of materials, propagation of light in optical medium and related phenomena of quantum mechanics. Some of these models arise in the study of quantum mechanical systems and nonlinear optics, while others have purely geometric origins. Better mathematical description of the properties of the solutions of these equations, especially their asymptotic behavior in time and space, will greatly improve our understanding of the properties of these nonlinear field theories and help in the development of technologies that use them.