Asymptotics of solutions for dispersive quasilinear problems

Brown University, Providence RI

Investigators

Abstract

This project aims to study several phenomena in the broad category of nonlinear dispersive problems. These problems arise from varied domains of physics, such as the interaction between the structure of space-time and matter in general relativity, the influence of changes in the topography on the propagation of ocean waves, or simply some toy models from plasma physics and laser propagation devised to understand the possible buildup of energy in smaller dispersive channels. These situations are different and yet all result from complex interactions between several different "modes of propagation" combining to create new dynamics that cannot simply be predicted by looking at each component of the system in isolation. A better understanding of these effects allows one then to develop better and simpler models that accurately describe the behavior of the system under consideration for large time. The objective of this project is to describe in detail examples where such effects are relevant and to highlight the underlying common mathematical structure, while at the same time isolating the specific features responsible for each particular phenomenology. The main theme of this project is the study of the long-time behavior of three different nonlinear hyperbolic and dispersive systems in cases where scattering does not hold and nonlinear effects have a strong impact on the qualitative behavior of solutions. The principal investigator will consider the following: (i) a problem from general relativity, namely, the stability of Minkowski space for the Einstein equations in the presence of a massive scalar field; (ii) a model (NLS) problem that involves the possible growth of Sobolev norms on domains of smaller volume; (iii) a model from water-wave theory that seeks to demonstrate how a sudden and localized change in the bottom topography influences the propagation of solitary waves. In each case the principal investigator will focus on controlling solutions globally using a combination of Fourier and harmonic analysis, bilinear estimates, ordinary differential equations and geometric methods (vector fields, normal forms, tools from Hamiltonian dynamics), and adapted function spaces. The objective is to derive improved "effective" dynamics that control asymptotic behavior.