Mathematical Aspects of Classical Fields

New York University, New York NY

Investigators

Abstract

This proposal is concerned with questions related to the behavior of solutions to nonlinear wave equations under perturbation. In particular we are concerned with geometric PDEs such as wave maps and Schrodinger maps. These equations can be viewed as constrained motion in space where the constraint forces the solution to lie on a subspace such as a manifold or a metric space. This leads to singularly perturbed wave and Schrodinger equations. We propose to study the relationship between the wave map equations and the constrained motion in terms of convergence and in terms of the dynamics of solutions. Specifically we propose to study: 1) Convergence of the constrained motion when the data are not "well prepared" leading to rapid oscillations; 2) the persistence of periodic, quasi-periodic, and homoclinic orbits that asymptote to quasi-periodic orbits; 3) wave maps into the Heisenberg Group, as a metric space; and finally 4) the relationship between Schrodinger maps and wave maps, where the wave map is considered as a constrained motion on a Lagrangian submanifold of the Schrodinger map into a Kahler manifold. The dynamics of many physical systems can be described by geometric evolution equations. For example dynamics of the magnetization field in a ferromagnetic material is described by the Landau-Lifshitz equation; the study of a vibrating membrane in a crystalline structure where there is a uniform distribution of dislocations, is described by wave equations into the Heisenberg Group (which also arises in other physical applications such as control theory, and the motion of robot arms). These geometric evolution models need not be exact, but should be considered as a limit of a singularly perturbed system. The general questions that we propose to study are: What relationships exist between solutions of the geometric equations and the singularly perturbed equations? How much of the rich dynamical structure (periodic, quasi-periodic, and breather type solutions) that the geometric equations have will persist under this type of singular perturbation? One of the major goals of our work is to develop analytical methods to deal with convergence questions related to this type of perturbations. Another important goal is to develop and use infinite dimensional dynamical systems techniques in conjunction with analytical methods to answer the persistence question and ultimately to bridge the gap between finite dimensional dynamical systems and PDEs.