Nonlinear Waves

New York University, New York NY

Investigators

Abstract

This project is concerned with the study of the evolution of free boundaries and interfaces in fluid flows and the impact of resonances on their evolution. The principal investigator will collaborate with P. Germain and N. Masmoudi to develop the space-time resonances method. This method consists of isolating frequencies that are resonant in time and located in the same region in space. In order to determine such frequencies, a detailed analysis of the wave packets and their interaction will be carried out. This will include a study of multilinear operators a la Coifman-Meyer, as well as operators with flag singularities. The project also incorporates the study of singular limits of modulated waves into the research. We anticipate that the limiting behavior will be determined by space-time resonant waves. All of this research will require a close integration of the geometric energy method (that was developed by the principal investigator jointly with C. Zeng) with the space-time resonance method described above. Many physical phenomena, such as surface ocean waves and their interaction with wind, and engineering problems, such as laser cooling to produce Bose-Einstein condensates, can be explained or solved by evoking the phenomena of resonances. Together with dispersion, resonances form the backbone of the analytical tools that have been developed to study the stability of nonlinear waves. The development of the space-time resonance method is a cornerstone of such an endeavor. It will bring together different areas of mathematics and applied mathematics (harmonic analysis, partial differential equations, dynamical systems, fluid dynamics, vortex dynamics), where new tools will be developed. It also brings existing methods to a new setting that extends their applicability far beyond what was previously possible. This project will also include the training graduate students. This training will be unique, since working on any of the projects will require considerable knowledge in different areas of mathematics and physics.