CAREER: Emerging Challenges in Wave Turbulence Theory

波浪湍流是描述远离热平衡的波浪之间非线性动力相互作用的一种普遍物理现象。波湍流的例子发生在经典的表面水波中，也发生在涉及玻色-爱因斯坦凝聚（BEC）的量子动力学系统中。尽管描述自然界中随机波相互作用过程的波场非常多样，但有一个共同的数学框架，通过与量子或经典波系统中弱非线性相互作用相关的概率密度来建模和描述谱能量转移的动力学。概率密度是波动方程（WK）的解，其非局部相互作用算子是动力学型的。该项目旨在通过研究相关的波浪湍流理论来解决非线性波浪理论中的公开挑战。该项目的一个组成部分是其教育部分和让各级学生参与研究的机会。为此，首席研究员将为代表性不足和残疾的K-12学生组织暑期学校，设计偏微分方程、波动湍流和统计物理学的研究生课程，并组织本科生和研究生卓越研究实习计划以及数学-物理会议为年轻研究人员举办。此外，主要研究者将参加NSF RTG SMU夏季本科生研究计划，参与者来自SMU和德克萨斯州格兰德河谷大学（西班牙裔服务机构）以及SMU汉密尔顿本科生研究学者和SMU本科生研究助理计划。这个项目集中在WT理论的三个主要主题。第一个主题是研究三波动力学方程的严格证明，用费曼图解法。 第二个主题是使用重整化群方法建立海洋Garrett-Munk谱的数学基础。第三个主题是有限温度BEC系统的解决方案的奇点的有限时间形成的证明，增加了一个新的，以前失踪，碰撞运营商得出的Yves Pomeau和首席研究员。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。

Wave turbulence (WT) is a general physical phenomenon describing the nonlinear dynamical interactions of waves far from thermal equilibrium. Examples of wave turbulence occur in classical surface water waves and also in quantum dynamical systems involving a Bose-Einstein condensate (BEC). Even though wave fields describing the processes of random wave interactions in nature are enormously diverse, there is a common mathematical framework that models and describes the dynamics of spectral energy transfer through probability densities associated with weakly non-linear interactions in quantum or classical wave systems. The probability densities are solutions of wave kinetic (WK) equations, whose nonlocal interaction operators are of kinetic type. This project aims to tackle open challenges in the theory of nonlinear waves via the study of the associated wave turbulence theory. An integral part of the project is its educational component and the opportunities to involve students from all levels in the research. To this end, the principal investigator will organize summer schools for underrepresented and disabled K-12 students, design graduate courses on Partial Differential Equations, Wave Turbulence and Statistical Physics, and organize an undergraduate and graduate research internship program of excellence and a mathematics-physics conference for young researchers. In addition, the principal investigator will participate in the NSF RTG SMU summer undergraduate research program, with participants from both SMU and Texas Rio Grande Valley University (a Hispanic-serving institution) as well as the SMU Hamilton Undergraduate Research Scholars and the SMU Undergraduate Research Assistant Programs. This project concentrates on three main topics in WT theory. The first topic is to study the rigorous justification of 3-wave kinetic equations, by the Feynman diagrammatic approach. The second topic is to establish a mathematical foundation for the Garrett-Munk spectrum of the ocean, using the renormalization group method. The third topic is the proof of the finite time formation of singularities of solutions to the finite temperature BEC system, with the addition of a new, previously missing, collision operator derived by Yves Pomeau and the principal investigator. Several ideas from kinetic theory, dispersive equations, oceanography, and quantum mechanics will be combined to study the proposed problems.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.