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

这个项目的主要科学目标是研究某些非线性发展偏微分方程组(PDE)和这些方程的自然起源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(玻色爱因斯坦凝聚体)到宏观(深海中的无赖波)，以及气体动力学(弗拉索夫方程)。为了应对研究这些方程式的重要挑战，投资促进机构采用了一种结合了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多粒子水平上识别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不同阶段进行合作，并为学生和博士后学者提供研究机会和支持。其他活动包括三个旨在培训、传播和促进进一步研究的年度研究讲习班。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。第一个方向涉及非线性发展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Ebin-Marsden的开创性计划的指导下，在流体力学的背景下关于这种推导的新观点。第二个方向以一维三次非线性薛定谔(NLS)方程的可积性为基础，分两条线进行研究。其中一个问题集中于通过一系列项目来探索一维立方NLS的可积性的起源，这些项目旨在揭示多粒子水平上的可积性的正确类似，然后证明这些类似物对应于NLS水平上的已知性质。第二条线索来自Lebowitz、Rose和Speer的工作，他们假设均衡行为的宏正则系综描述对于可积偏微分方程组是错误的。PIS计划通过构建合适的替代品来解决这一重大悬而未决的问题。这一奖项反映了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。

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.