Probabilistic Aspects of Dispersive and Wave Equations

几个世纪以来，偏微分方程（PDE）在理解物理和自然现象方面发挥了重要作用。色散/波动方程模拟自然界中普遍存在的波传播现象。它们还描述了量子物理学的基本定律，量子物理学是20世纪最伟大的成就之一。本课题通过引入概率论的思想，研究色散方程和波动方程的基本问题。该项目的研究成果将推动波浪湍流数学理论的发展，该理论在等离子体物理、非线性光学和海洋学等领域具有重要的应用，并在量子场论和统计物理中发挥关键作用的哈密顿系统的吉布斯测度分析。由于其范围和与物理和科学的联系，该项目还将促进跨学科的互动。作为该项目的一部分，首席研究员（PI）正在培训初级研究人员，并为维护南加州大学STEM学科的多样性做出贡献。该奖项支持5个研究项目（A-E）。前三个项目涉及波浪湍流的数学理论。在A项目中，PI将波动动力学方程的短动力学时间推导扩展到更长的动力学时间。这是理论发展的重要一步，因为它超越了摄动状态，也将揭示玻尔兹曼方程长期推导的长期开放问题。在项目B中，PI计划推广这一推导，以涵盖所有推测的标度定律，这在物理上是很好的动机，也会导致新的数学上有趣的结构。在物理学文献中没有出现的新的重要组合结构和消去有望被发现。项目C考虑的是水波的波动湍流问题，这一问题自20世纪60年代以来一直由物理学家研究。在数学上，它是一个拟线性方程，需要大量的新思想才能得到与半线性方程相似的结果。最后两个项目涉及吉布斯和其他统计物理和量子场论中的不变测度。项目D涉及二维双曲正弦-戈登方程的吉布斯测度，这是一个包含近临界情况的重要模型。在这里，目标是进一步发展PI在早期工作中引入的随机张量理论。E项目通过概率论和可积系统技术的结合，研究一维三次非线性Schrödinger方程的白噪声测量的不变性，这是至关重要的，但也是完全可积的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

For centuries, partial differential equations (PDE) have played a fundamental role in understanding physical and natural phenomena. Dispersive/wave equations model wave propagation phenomena which are ubiquitous in nature. They also describe the basic laws of quantum physics, which is one of the greatest achievements of the 20th century. This project studies fundamental questions about dispersive and wave equations by introducing ideas from probability theory. The results of the project will advance the mathematical theory of wave turbulence, which has important applications to plasma physics, nonlinear optics, and oceanography, and the analysis of Gibbs measures for Hamiltonian systems, which plays a key role in quantum field theory and statistical physics. Due to its scope and connections to physics and science, the project will also promote interdisciplinary interactions. As part of the project, the Principal Investigator (PI) is training junior researchers and contributes to maintaining the diversity in STEM disciplines at University of Southern California.This award supports work on five research projects (A-E). The first three projects are concerned with the mathematical theory of wave turbulence. In Project A, the PI extends the short kinetic time derivation of wave kinetic equation to longer kinetic times. This is a major step in the development of the theory, as it goes beyond the perturbative regime and will also shed light on the longstanding open problem of the long-time derivation of the Boltzmann equation. In Project B, the PI plans to generalize this derivation to cover the full range of conjectured scaling laws, which is physically well motivated and also leads to new mathematically interesting structures. New significant combinatorial structures and cancellations which are not present in the physics literature are expected to be discovered. Project C considers the wave turbulence problem for water waves, which has been studied since the 1960s by physicists. Mathematically, it is a quasilinear equation and substantial new ideas are required to obtain results similar to the ones available in the semilinear case. The last two projects concern Gibbs and other invariant measures in statistical physics and quantum field theory. Project D concerns the Gibbs measure for the 2D hyperbolic sine-Gordon equation, which is an important model that contains near-critical scenarios. Here, the goal is to further develop the random tensor theory introduced by the PI in earlier work. Project E investigates, through a combination of techniques from probability theory and integrable systems, the invariance of the white noise measure for the one-dimensional cubic nonlinear Schrödinger equation, which is critical but also completely integrable.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.