Integrable and Non-Integrable Dispersive Partial Differential Equations

可积和不可积色散偏微分方程

• 批准号: 1763074
• 负责人: Monica Visan
• 金额: $ 27万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1763074&HistoricalAwards=false
• 关键词: Integrable; Non; Dispersive; Partial; Differential

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.

PI提议研究的模型之一是Korteweg-De Vries（KDV）方程。 该方程的得出比一百年前得出的是解释浅水通道中长波的行为。 在1960年代，普林斯顿血浆物理实验室的研究人员表明，该方程式展示了许多新颖的特征，这激发了数学家和物理学家的兴趣。 但是，尽管多年来受到了所有关注，但PI和她的合作者最近才证明了最少假设的解决方案的存在。 他们的工作中的一种成分是最近发现了新的保护法。 该项目概述了现在可以使用此发现攻击的其他几个问题。 该项目背后的另一个主要动力是证明复杂的瞬态动力学在遥远的未来将其置于简单的动态中。 这种现象的物理意义取决于其在扰动下的稳定性。 在过去，PI调查了方程式的确定性扰动，当前项目通过在存在（随机）噪声的情况下考虑稳定性来将该主题朝一个新的方向发展。 该项目的重点是在非线性分散偏微分方程，完全可以集成的系统和随机部分微分方程的交点上的几个问题。 PI发现针对KDV的新显微镜保护定律为治疗三个看似无关的关于KDV的长期兴趣的问题打开了大门：最佳的规律性适应性良好，符号性的非斜率和白噪声的不变性。 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应揭示Landau-Lifshitz和Cutic NLS模型的物理重态化，这些模型将确保该数据的适当性。认为值得通过基金会的智力优点和更广泛影响的评论标准来评估值得支持。

项目成果

Invariant Measures for Integrable Spin Chains and an Integrable Discrete Nonlinear Schrödinger Equation

可积自旋链的不变测度和可积离散非线性薛定谔方程

• DOI: 10.1137/19m1265314
• 发表时间: 2020
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Angelopoulos, Yannis;Killip, Rowan;Visan, Monica
• 通讯作者: Visan, Monica

Breakdown of Regularity of Scattering for Mass-Subcritical NLS

质量亚临界NLS散射规律的分解

• DOI: 10.1093/imrn/rnaa072
• 发表时间: 2020
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Lee, Gyu Eun

Global Well-Posedness for the Fifth-Order KdV Equation in $$H^{-1}(\pmb {\mathbb {R}})$$

• DOI: 10.1007/s40818-021-00111-4
• 发表时间: 2019-12
• 期刊: Annals of PDE
• 影响因子: 2.8
• 作者: Bjoern Bringmann;R. Killip;M. Vişan

Invariance of white noise for KdV on the line

• DOI: 10.1007/s00222-020-00964-9
• 发表时间: 2019-04
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: R. Killip;Jason Murphy;M. Vişan

Sonin's argument, the shape of solitons, and the most stably singular matrix

• 发表时间: 2018-11
• 期刊: arXiv: Analysis of PDEs
• 作者: R. Killip;M. Vişan