This project focuses mainly on analyzing the long-time behavior of certain dispersive partial differential equations. All the equations considered in the project have a physical origins: the Schrodinger maps equation is known as the Heisenberg model in ferro-magnetism, the spin-models have a similar origin, while the Zakharov system comes from plasma physics. From a mathematical point of view, most of the problems the principal investigator intends to address lie at the cutting-edge of research in partial differential equations. The dynamics of various equations with large data is a very important problem in the field. Some major breakthroughs have been achieved during the past few years, and the analysis involved is highly nontrivial. Understating the impact of the geometry of the target manifold on the evolution of the equation (in the case of Schrodinger maps and spin-models) is of great interest. Research into systems that lack scaling (like the Zakharov system) is extremely challenging, especially in the absence of usable conservation laws. While, strictly speaking, the problems to be explored in this project belong to the field of partial differential equations, the research requires the use of fine tools from other areas of mathematics, notably harmonic analysis and Riemannian geometry.
One of the main reasons that mathematics is useful to the broad scientific community, and in turn to society as a whole, is that it provides one of the most rigorous frameworks for constructing theories that explain the world around us and predict future events. The field of partial differential equations is, to a great extent, the study of models arising from physics. Everyone is aware of the existence of light, heat, fluid flow, magnetism, etc. These are all natural phenomena that, once they are well understood, can lead to major discoveries whose impact on human lives is tremendous. A scientific approach to the study of a natural phenomenon follows a standard pattern. One investigates the complexity of the phenomenon, determines its essential features, and writes down a differential equation that describes the evolution in time of the object under study. Next, one studies the long-time behavior of the mathematical model and describes its evolution in qualitative terms. (In the event that this process identifies potential singularities, a prime interest of the principal investigator, then the phenomenon becomes directly relevant to the current project.) Finally, if the mathematical analysis agrees with the empirical observation of the phenomenon, then the mathematical model is validated, which often opens a wide range of possible applications. On the other hand, if discrepancies arise between the mathematics and the empirical observations, then one seeks to refine the mathematical model, usually by allowing for greater complexity. The new model undergoes a similar mathematical analysis and so on, until a good mathematical model matching the physical reality is found.
