FRG: Collaborative Research: New Challenges in the Derivation and Dynamics of Quantum Systems

该项目的主要科学目标是研究某些非线性演化偏微分方程（PDE）与推导这些方程的自然祖粒子系统之间的相互作用。所考虑的方程是波传播现象的基本模型，范围从微观（玻色爱因斯坦凝聚）到宏观（深海中的异常波），以及气体动力学（Vlasov方程）。为了解决研究这些方程的重要挑战，pi采用了一种结合确定性和概率观点的创新方法。根据这些PDE的定性性质，本项目的主要目标是在许多粒子水平上确定这些性质的正确类似物，并证明这些性质与PDE水平上的已知性质相对应。该奖项将促进美国研究人员在其职业生涯的各个阶段之间的合作，并为学生和博士后学者提供研究机会和支持。其他活动包括三个年度研究讲习班，目的是培训、传播和鼓励进一步的研究。在他们的分析中，pi考虑了数学物理前沿的两个不同但密切相关的研究方向，非线性偏微分方程和概率论。第一个方向涉及非线性演化方程的哈密顿结构的推导，包括动力学方程，如弗拉索夫方程，以及在流体力学背景下由Ebin-Marsden的开创性程序指导的关于这些推导的新观点。第二个方向是基于一维三次非线性薛定谔方程的可积性，并追求两条探究线。其中一个问题是通过一系列旨在揭示多粒子水平上可积性的正确类似物的项目来探索1D立方NLS可积性的起源，然后证明这些与NLS水平上的已知性质相对应。第二条探究线源于Lebowitz， Rose和Speer的工作，他们假设对于可积偏微分方程，平衡行为的大规范系综描述预计是错误的。ppi计划通过构建一个合适的替代品来解决这个重大的开放性问题。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。

The main scientific goal of this project is to study the interplay between certain nonlinear evolution partial differential equations (PDE) and the natural progenitor particle systems from which these equations are derived. The equations considered are fundamental models for wave propagation phenomena ranging from the microscopic (Bose Einstein Condensate) to the macroscopic (rogue waves in deep sea), and for the dynamics of gases (Vlasov equation). To address important challenges in studying these equations, the PIs adopt an innovative approach combining deterministic and probabilistic perspectives. Informed by the qualitative properties of these PDE, the principal objective of this project is to identify the correct analogues of such properties at the many particle level, and to demonstrate that these correspond to the known properties at the PDE level. The award will foster collaborations among US based researchers at various stages of their careers and provide research opportunities and support for students and postdoctoral scholars. Additional activities include three annual research workshops aimed at training, dissemination, and stimulate further research. In their analysis, the PIs consider two different but intimately related research directions at the forefront of mathematical physics, nonlinear PDE and probability. The first direction concerns the derivation of the Hamiltonian structure for nonlinear evolution equations, including kinetic equations such as the Vlasov equation, as well as a novel viewpoint on such derivations guided by Ebin-Marsden's seminal program in the context of hydrodynamics. The second direction is rooted on the integrability of the 1D cubic nonlinear Schrodinger (NLS) equation and pursues two lines of inquiry. One of these questions focuses on exploring the origins of integrability of the 1D cubic NLS through a series of projects aimed at unveiling correct analogues of integrability at the many particle level, and then at demonstrating that these correspond to the known properties at the NLS level. A second line of inquiry stems from the work of Lebowitz, Rose and Speer who posited that the grand canonical ensemble description of equilibrium behavior is expected to be false for integrable PDE. The PIs plan to settle this major open problem by constructing a suitable substitute.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.