Quantified dynamics of nonlinear dispersive PDE

Brown University, Providence RI

Investigators

Abstract

This project will study nonlinear dispersive equations, with particular emphasis on the nonlinear Schroedinger equation, Korteweg-de Vries equation, and nonlinear Klein-Gordon equation. New techniques have emerged for studying the dynamics of solitons and finite-time blow-up solutions, such as local virial estimates, new Lyapunov functionals, and concentration compactness machinery. The principal investigator will seek to implement and develop these techniques further in the investigation of several scientifically relevant problems. The dynamics of solitary waves in the presence of an external potential or under the influence of a time-dependent nonlinear coefficient for single and multiple bright and dark solitons will be considered. The ability of a strong confining potential in one direction to produce an effective reduction in dimensionality will be explored. New criteria on initial data for predicting finite-time blow-up of solutions will be developed, and the PI will seek to construct new types of singular ring blow-up solutions recently observed numerically. Long-time behavior of global solutions will also be studied. The equations studied in this project arise as important physical models. Solitons appear as well-localized stable structures, while blow-up is associated with a sharp focusing of a wave and ultimate break-down of the physical model. For example, the nonlinear Schroedinger equation is the wave function for a collection of ultracold atoms in a Bose-Einstein condensate. Since its Nobel-prize winning realization in the laboratory in 1995, further experiments have produced solitons and finite-time blow-up, and there is now a large body of physics literature motivating many of the problems the principal investigator intends to consider. The problems proposed bring together physicists, numerical analysts, and pure mathematicians. They also serve as an excellent educational tool, since they can be tailored to students at the undergraduate and graduate level. To stimulate interest among such students, the principal investigator will give talks in the undergraduate seminar and teach an advanced topics course at Brown University. Web demonstrations of this research could be produced, for example, through a summer undergraduate project funded by this grant.