Waves are ubiquitous in nature. They are central in describing fundamental physical phenomena at all scales, from quantum mechanics to general relativity. When in a given physical system a large number of interacting waves are present, the description of an individual wave is neither possible nor relevant. What becomes of physical importance and practical use is the density and statistics of the interacting waves. This is Wave Turbulence Theory, which has the remarkable feature of universally predicting the evolution of the wave action spectral density of interacting wave systems. This prediction is accomplished through the Wave Kinetic Equation, which is an evolution equation for the wave action density and it is the wave analogue of the Boltzmann kinetic equations for particle interactions. This equation has been successfully applied to describe waves in the ocean (gravity, internal, and capillary waves), and waves in magnetized fluids (solar winds, interstellar media, fusion plasmas). Many new applications have recently emerged in fields like condensed matter (Helium superfluids, Bose-Einstein condensates), nonlinear optics, and, most recently, in the study of gravitational waves in the early universe.
A perfect example to illustrate the importance of Wave Turbulence Theory, is that of forecasting surface gravity waves in the oceans. Nowadays, the Wave Kinetic Equation is the standard tool for performing operationally the forecasting of surface gravity waves in the oceans. Everyday, the Wave Kinetic Equation, with forcing provided by meteorological models, is numerically integrated and an output, in terms of integral quantities of the wave spectrum, is usually released every three hours, with a spatial resolution of about 14 km. Wave forecasting is fundamental for navigation, and safety operations at sea and on off-shore platforms.
Although the Wave Kinetic Equation has been widely used, its range of applicability has never been put on a rigorous mathematical foundation. Its predictions of the energy spectrum are not always in agreement with empirical data. This discrepancy could, in part, be explained by the lack of a rigorous mathematical theory. This collaboration is the first attempt for a systematic coordinated study of Wave Turbulence Theory in a large-scale project, bringing together state-of-the-art skills in the areas of mathematics and physics, with theoretical, experimental, and numerical expertise. It is a joint effort of several groups of researchers who are ready to collectively collaborate, question all assumptions and approximations, and coordinate the progress on an interdisciplinary set of problems.