Nonlinear Dispersive Waves

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The main objective of this project is to improve our understanding of the solutions for a broad array of partial differential equations that can be all described as Nonlinear Waves. These equations arise as models for physical phenomena, which play important roles in our world and sometimes even in our daily lives. Key examples range from the water waves at the surface of our oceans, the quantum mechanical interaction of multiparticle systems, to the surface dynamics of gaseous stars. The common feature in all of these phenomena is the interplay between linear waves and nonlinear interactions, whose balance affects both short and long range dynamics. Different parts of the project aim to study both short time phenomena, such as singularity formation, as well as their long time behavior, such as scattering. The project provides research training opportunities for graduate students. The proposed research spans a broad array of research topics in nonlinear partial differential equations. The problems to be investigated are all associated with the field of nonlinear dispersive equations, but also carry strong connections to related areas such as geometry, harmonic analysis, complex analysis and microlocal analysis. For the most part, these problems are strongly nonlinear, and also fundamentally based on physical models from areas such as fluid dynamics, electromagnetism, and general relativity. In a nutshell, one can view the objective of this work to be the understanding of nonlinear wave interactions, beginning with short time scales, continuing with long time scales, all the way to scattering and blow-up phenomena. One of the main research areas targeted by this project is in fluid dynamics, more precisely water waves, as well as a broader class of free boundary problems. This has been an area of intense interest in recent years, not in the least because of its great potential for applications, ranging from oceanography to medical science and to stellar dynamics. But the most interesting problems are still unresolved, and their study is bringing forth an array of extremely difficult questions. Another goal is the study of completely integrable systems, which often arise as models for more difficult nonlinear problems in fluid dynamics in general, and water waves in particular. The main objective is to gain a sufficiently accurate understanding of the long time dynamics, which combine solitons, dispersive shocks and nonlinear scattering in a complex pattern. The study of geometric nonlinear wave equations is a third objective of this project. After the recent proof of the Threshold Conjecture for Wave Maps and Yang-Mills systems, the current work is directed toward a full classification of blow-up solutions and of non-scattering solutions, as predicted by the Soliton Resolution Conjecture. Finally, quasilinear wave and Schrodinger evolutions bring forth some of the most challenging problems in nonlinear partial differential equations, and understanding both their short and long time dynamics is another major goal of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.