Many descriptions of real-life processes lead to the formulation of equations whose solutions change over time. For example, wave- or dispersive-type equations display this kind of behavior. These evolution equations provide the possibility of constructing solutions from prescribed initial conditions. In some evolution processes, a prescribed restriction (or boundedness) over time is given and the question is then whether the boundedness property persists forever or whether some sort of "unboundedness" can arise (e.g., a freak wave suddenly appearing in the ocean or a self-focusing burn in laser optics). These evolution processes are fundamental in nature, yet we are still in the infancy of their full analytical description. This project seeks to shed new light on the global behavior of solutions to nonlinear evolution partial differential equations (PDE), in particular, of dispersive PDE for which the nonlinearities cause a significant difference in global behavior compared with linear evolution. A special emphasis will be placed on the study of singularity formation, such as collapse and blow-up. The principal investigator's research program starts from advancing the theory of collapse for the simplest models in nonlinear wave equations, proceeds by expanding this theory to more generalized models and equations, and then goes on to explore the dynamics of solutions for simple dispersive systems, which have hitherto been studied little but which describe important models in the physical world. The ultimate goal of the research is to develop singularity theory for dispersive systems like the Davey-Stewartson system. These equations arise in physical contexts such as the description of water waves or acoustic waves, and in various fields such as laser optics and fluid and air dynamics.
This project will advance the frontiers in the theory of nonlinear evolution equations, in particular, improving our understanding of collapse phenomena. It will produce a rigorous analytical description of concentration and aggregation for physical applications in real life. This work will be enhanced by collaborations at the national and international levels and will help strengthen interinstitutional ties, for example, between the George Washington University (GWU) and Howard University. In addition, the project will provide mathematical training and educational experiences at all mentoring levels, reinforcing future US competitiveness. It will focus on attracting, training, and retaining in mathematics and science outstanding female students and students from underrepresented groups. Vertical integration of educational activities, from middle school to the graduate level and beyond, will help to reinforce the project's agenda and contribute to the recruitment and retention of skillful workforce in the STEM fields. These activities will occur at the middle school level through the DC Math Circle program; at the high school level through the GWU summer precollege program (which annually recruits over two hundred students nationally); at the undergraduate level by engaging the principal investigator in the supervision of senior research theses and her serving as a mentor and guest lecturer at the GWU "Summer Program for Women in Math" ; and, finally, at the graduate level by organizing the DC Grad Camp, with the emphasis on attracting female and underrepresented minority students.