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）与这些方程所源自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，从微观（玻色爱因斯坦凝聚）到宏观（深海中的流氓波），以及气体的动力学（弗拉索夫方程）。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。通过这些PDE的定性性质，本项目的主要目标是在许多粒子水平上识别这些性质的正确类似物，并证明这些对应于PDE水平上的已知性质。该奖项将促进美国研究人员在其职业生涯的各个阶段的合作，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，旨在培训、传播和促进进一步研究。 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PDE和概率。第一个方向涉及非线性发展方程（包括弗拉索夫方程等动力学方程）的汉密尔顿结构的推导，以及由Ebin-Marsden在流体动力学背景下的开创性计划指导的关于此类推导的新观点。第二个方向是植根于1D立方非线性薛定谔（NLS）方程的可积性，并追求两条调查线。其中一个问题的重点是通过一系列旨在揭示许多粒子水平上可积性的正确类似物的项目来探索一维立方NLS可积性的起源，然后证明这些对应于NLS水平上的已知性质。第二条调查线源于Lebowitz，Rose和Speer的工作，他们假设平衡行为的巨正则系综描述对于可积PDE是错误的。PI计划通过构建一个合适的替代品来解决这个主要的开放问题。该奖项反映了NSF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.