Collaborative Research: On New Directions for the Derivation of Wave Kinetic Equations

The beauty and power of mathematics is to recognize common features in a variety of phenomena that may look physically different. This is certainly the case when one studies wave turbulence theory. This theory is focused on the fundamental concept that 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 important and practical is the description of the density and the statistics of the interacting waves. Arguably the most recognizable and fundamental objects within this theory are the wave kinetic equations. These equations, their solutions and their approximations have been used to study a variety of phenomena: water surface gravity and capillary waves, inertial waves due to rotation and internal waves on density stratifications, which are important in the study of planetary atmospheres and oceans; Alfvén wave turbulence in solar wind; planetary Rossby waves, which are important for the study of weather and climate evolutions; waves in Bose-Einstein condensates (BECs) and in nonlinear optics; waves in plasmas of fusion devices; and many others. This project will tackle foundational questions in wave turbulence theory through rigorous mathematical analysis. In addition, the project will promote collaborations, facilitate the dissemination of interdisciplinary research, and provide opportunities for undergraduate and graduate students to work on a multifaceted and forward-looking line of mathematical research. This project tackles challenging problems at the intersection of the physics and the mathematical analysis of nonlinear interactions of waves that are central in the study of wave turbulence theory. These problems include the rigorous derivation of wave kinetic equations, the analysis of the 4-wave kinetic equation for the Fermi- Pasta-Ulam-Tsingou (FPUT) chain and the well-posedness of a geometric wave equation via Feynman diagrams in the energy regime. The research proposed will not just address important open problems but will contribute to the interdisciplinary development of several new and complex tools both in mathematics and physics. The proposal aims at providing these tools by blending Feynman diagrams, harmonic analysis, probability, combinatorics, incidence geometry, kinetic theory, dispersive PDE, quantum field theory and the FPUT chain. 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.