The main thrust of this project is to further the understanding of the long-time behavior of solutions to dispersive equations with broken symmetries. Specifically, the principal investigator considers global well-posedness, scattering, soliton formation, and finite-time blowup questions for equations such as the following: nonlinear Schrodinger outside nontrapping domains, Gross-Pitaevskii, nonlinear Schrodinger with combined focusing and defocusing nonlinearities, generalized Korteweg-de Vries. The common feature of these equations is that each has one or more broken symmetries such as spatial-translation, scaling, or Galilei invariance. By this is meant that, while the solution can concentrate at arbitrary positions in space (or length scales or frequency locations), the evolution depends nontrivially on the position (or scale or frequency location) of concentration. One of the remarkable new phenomena that can occur in this setting is that minimizing sequences of solutions to an equation with broken symmetries can converge to a solution of the same equation but in a different geometry, or even to a solution of an entirely different equation.
The partial differential equations to be investigated in this project arise as effective models in various areas of physics, such as models for waves in shallow water, laser light in Kerr media, Bose-Einstein gases in a trap, and superconductors. While they are oversimplified models for engineering purposes, the study of these examples allows one to focus on the new phenomenology that arises when symmetries are broken. Stability of results under such perturbations is essential to make a meaningful connection to real-world systems; indeed, engineering limitations mean that real-world systems never have perfect symmetry. Conversely, failure of stability implies that computer simulations cannot be relied upon to accurately predict experimental phenomena.
