The Principal Investigator will continue work on the regularity properties of certain nonlinear dispersive PDE evolving on either compact domains with periodic boundary conditions, outside of an obstacle in space, or in all of space. Examples of the equations, which the project will study, include KdV, certain nonlinear Schroedinger equations, and nonlinear second order hyperbolic equations.
Many physical theories - e.g. fluid dynamics, elasticity, and gravitation - culminate in mathematical equations whose solutions cannot be recovered precisely by any known formulae. The PI's research takes two routes to a more qualitative understanding of such problems. One approach aims to understand the complicated system as a perturbation of a simple, but related theory. A complementary approach views the complicated solution as a sum of simple waves of constant frequency, focusing on the interaction of these constituents as time passes. The goal of this analysis is to uncover unrecognized physical phenomena, and to design better computational models of the original physical theory.
