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

数学的美丽和力量在于识别各种现象中的共同特征，这些现象可能看起来物理上不同。当人们研究波动湍流理论时，情况肯定是这样的。这个理论的重点是基本概念，即当在一个给定的物理系统中存在大量相互作用的波时，单个波的描述既不可能也不相关。 变得重要和实用的是相互作用波的密度和统计的描述。可以说，在这个理论中，最容易辨认和最基本的对象是波动动力学方程。这些方程及其解和近似值已被用于研究各种现象：水面重力波和毛细波、旋转引起的惯性波和密度分层的内波，这些对研究行星大气和海洋很重要;太阳风中的阿尔文波湍流;行星Rossby波，这些对研究天气和气候演变很重要;玻色-爱因斯坦凝聚体（BEC）和非线性光学中的波;聚变装置等离子体中的波;以及许多其他方面。这个项目将通过严格的数学分析来解决波浪湍流理论中的基础问题。此外，该项目还将促进合作，促进跨学科研究的传播，并为本科生和研究生提供机会，从事多方面的和前瞻性的数学研究。该项目解决了在波浪湍流理论研究中处于核心地位的波浪非线性相互作用的物理和数学分析交叉点的挑战性问题。这些问题包括严格推导波动力学方程，分析Fermi-Pasta-Ulam-Tsingou（FPUT）链的四波动力学方程，以及通过Feynman图证明几何波方程在能区的适定性。这项研究不仅将解决重要的开放性问题，而且将有助于数学和物理学中几个新的复杂工具的跨学科发展。该计划旨在通过融合费恩曼图、谐波分析、概率论、组合学、入射几何、动力学理论、色散偏微分方程、量子场论和FPUT链来提供这些工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.