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CAREER: Emerging Challenges in Wave Turbulence Theory

职业：波浪湍流理论中的新挑战

基本信息

批准号：
2303146
负责人：
Minh-Binh Tran
金额：
$ 43万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-12-01 至 2026-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303146&HistoricalAwards=false
关键词：
CAREER Emerging Challenges Wave Turbulence

项目摘要

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.

波浪湍流是描述远离热平衡的波浪之间的非线性动力学相互作用的一种普遍物理现象。波动湍流的例子出现在经典的表面水波和涉及玻色-爱因斯坦凝聚体(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是非常不同的，但有一个共同的数学框架来建模和描述通过与量子或经典波系统中的弱非线性相互作用相关的概率密度来传递光谱能量的动力学。概率密度是波动动力学(WK)方程的解，其非局域相互作用算符是动态型的。该项目旨在通过对伴生波浪湍流理论的研究来解决非线性波浪理论中的开放挑战。该项目的一个组成部分是它的教育部分，以及让各个层次的学生参与研究的机会。为此，首席调查员将为代表不足和残疾的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的研究生课程，并组织本科生和研究生的优秀实习计划和青年研究人员的数学-物理会议。此外，首席研究员将参加NSF RTG SMU暑期本科生研究计划，参与者来自SMU和德克萨斯州里奥格兰德山谷大学(一所拉美裔服务机构)，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。这个项目集中在小波理论的三个主要主题上。第一个主题是用费曼图解方法研究三波运动方程的严格证明。第二个主题是利用重整化群方法为海洋的Garrett-Munk谱建立一个数学基础。第三个主题是证明有限温度BEC系统解的奇性的有限时间形成，增加了由Yves Pomeau和主要研究者导出的一个新的、先前丢失的碰撞算子。来自动力学理论、色散方程、海洋学和量子力学的几个想法将被结合起来研究提出的问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。