This project pursues several avenues of research in the theory of evolution partial differential equations. The principal investigator will study global behavior of solutions to nonlinear dispersive equations. These equations describe the evolution of waves, in particular in situations where different groups of waves propagate at different velocities so that waves "disperse" over time. Typical equations are the nonlinear Schrodinger and wave equations, since mathematically they represent the simplest nonlinear dispersive models and physically they describe wave propagation in applications such as nonlinear optics, hydrodynamics, plasma physics, quantum mechanics, and Bose-Einstein condensates. The nonlinear terms in dispersive equations create complicated but fascinating dynamics such as moving and interacting solitons, contracting or fixed singularities, collapses, etc. The study of solutions exhibiting such properties will be the focus of this proposal. The phenomenon of collapse is arguably "the most" nonlinear occurrence possible in an evolution equation, so it is, in general, very difficult to study. Recently there have been significant analytical strides made in the understanding of singular solutions to dispersive equations, hence it is quite timely to focus a research effort in this direction. Thus, the thrust of the project will be to provide descriptions of both the formation and the dynamics of singularities and to investigate the various thresholds that separate different types of global behavior in solutions to nonlinear dispersive equations.
This project studies differential equations that describe dispersion of waves appearing in various physical contexts: laser optics, fluid and air dynamics, and quantum and plasma physics, to name just a few. The methods developed here can be applied to problems not only in mathematics and physics, but also in statistics, engineering, biology, and medicine (for example,to study phenomena as wide-ranging as the formation of tsunamis or errant cardiac rhythms). Understanding when and how singularities occur is important for numerical simulations and real-time engineering applications as well. The principal investigator will disseminate the results of this project online, through a (freely accessible) institutional website and various free preprint servers, and she will present the latest advances at national and international conferences. Furthermore, this proposal includes mentoring and training activities at the highest levels of scientific and mathematical competency. The principal investigator will organize seminars, colloquia, undergraduate talks, and similar activities specifically designed to instruct students on the topics of the project, as well as on the general areas of partial differential equations and analysis. Her participation in the summer "Research Experiences for Undergraduates" programs organized by her home institution and nearby universities will be enhanced by this project. In particular, she will extend and develop her mini-courses and guest lectures at the "Summer Program for Women in Math" at her home institution in order to encourage women to pursue careers in mathematics and science.
