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.
Dispersive waves occur in a variety of physical systems such as nonlinear optics, atmosphere and ocean waves, and plasmas. Often these waves exhibit resonant interactions, i.e., interactions of waves that oscillate at the same frequency. Resonant interactions are of paramount importance in the long-time behavior of the waves present in the relevant physical system. Resonances can cause the wave amplitude to grow or can lead to genuine nonlinear behavior of solutions. It quantifies the long-time effects of the nonlinearities present in the problem. In addition to resonance, dispersion (waves moving into different parts of space) plays a central role in the long-time behavior of solutions. Dispersion often tames the effects of resonances by pushing resonant waves into different regions in space. It provides a mechanism to control the amplitude of resonant waves. Thus, together with dispersion, resonances form the backbone of the analytical tools which have been developed to study stability of nonlinear waves. The development of the space-time resonance method is a corner stone in such an endeavor. It brings existing methods together in a new setting that extends their applicability far beyond what was previously possible. This method is still in development and has been extended far beyond its initial inception. Nevertheless, there is lots of progress that needs to be made. An outcome of this award was the development of an algorithmic method to resonant interactions of weakly nonlinear waves. This procedure, namely, the space time resonance method has become the standard procedure to obtain long time behavior of weakly nonlinear waves. It helped to solve several outstanding problems in fluid flows and nonlinear wave equations.
