Integrable and Non-Integrable Dispersive Partial Differential Equations

PI提议研究的模型之一是Korteweg-de Vries （KdV）方程。这个方程是一百多年前推导出来的，用来解释长波在浅水通道中的行为。在20世纪60年代，普林斯顿等离子体物理实验室的研究人员证明，这个方程显示出大量的新特征，这引起了数学家和物理学家的兴趣。然而，尽管多年来一直受到关注，但在最小假设下的解的存在性直到最近才被PI和她的合作者证明。他们工作的一个组成部分是最近发现的新的守恒定律。这个项目概述了现在可以利用这一发现来解决的几个附加问题。这个项目背后的另一个主要推动力是证明在遥远的未来，复杂的瞬态动力学可以分解为简单的动力学。这种现象的物理意义取决于它在扰动下的稳定性。虽然在过去，PI研究了方程的确定性扰动，但当前的项目通过考虑（随机）噪声存在下的稳定性，将这一主题带到了一个新的方向。该项目主要研究非线性色散偏微分方程、完全可积系统和随机偏微分方程的交叉问题。PI对KdV的新微观守恒定律的发现，打开了一扇门，使我们可以处理关于KdV的三个长期存在的看似无关的问题：最佳正则性、适定性、辛非压缩和白噪声的不变性。此外，PI还提出了建立Landau-Lifshitz模型的Gibbs测度不变性和聚焦三次非线性Schr\ odinger方程（NLS）的白噪声不变性的连贯计划。这个程序包括建立与这些问题相关的物理原子模型的类似语句（PI已经成功地完成了），然后对相应的粗糙数据取连续体极限。这将揭示Landau-Lifshitz和立方NLS模型的物理重整化，以确保这些数据的适位性。由于朗道-利夫希茨模型的吉布斯测度对应于球体上的布朗运动，从纯概率的角度来看，这个问题也很有趣，因为它在这些路径上产生了保持哈密顿测度的流。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。

One of the models the PI is proposing to investigate is the Korteweg-de Vries (KdV) equation. This equation was derived more that a hundred years ago to explain the behavior of long waves in channels of shallow water. In the 1960s, researchers at Princeton's Plasma Physics Laboratory demonstrated that this equation exhibits a wealth of novel features, which have sparked the interest of mathematicians and physicists alike. However, despite all the attention it has received over the years, existence of solutions under minimal assumptions has been proved only recently by the PI and her collaborators. One ingredient in their work is the recent discovery of new conservation laws. This project outlines several additional problems that can now be attacked using this discovery. Another major impetus behind this project is to prove that complicated transient dynamics resolve into simple dynamics in the distant future. The physical significance of this phenomenon relies on its stability under perturbations. While in the past, the PI has investigated deterministic perturbations to the equations, the current project takes this theme in a new direction by considering stability in the presence of (random) noise. The project focuses on several problems that lie at the intersection of nonlinear dispersive partial differential equations, completely integrable systems, and stochastic partial differential equations. The PI's discovery of new microscopic conservation laws for KdV has opened the door to treating three seemingly unrelated problems of long-standing interest regarding KdV on the line: optimal regularity well-posedness, symplectic non-squeezing, and invariance of white noise. In addition, the PI is proposing a coherent plan for establishing invariance of the Gibbs measure for the Landau-Lifshitz model and invariance of white noise for the focusing cubic Nonlinear Schr\"odinger Equation (NLS). This program involves establishing the analogous statements for the physical atomic models associated with these problems (which the PI has successfully completed) and then taking the continuum limit for the corresponding rough data. This should reveal the physical renormalizations for the Landau-Lifshitz and the cubic NLS models that would ensure well-posedness for such data. As the Gibbs measure for the Landau-Lifshitz model corresponds to Brownian motion on the sphere, this problem is also interesting from a purely probabilistic point of view as yielding a Hamiltonian measure-preserving flow on such paths.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.