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建议研究的模型之一是Korteweg-de弗里斯（KdV）方程。 这个方程是一百多年前推导出来的，用来解释浅水中长波的行为。 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了这个方程具有丰富的新颖特征，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，但直到最近PI和她的合作者才证明了在最小假设下解的存在性。 他们工作的一个要素是最近发现的新守恒定律。 这个项目概述了几个额外的问题，现在可以使用这个发现攻击。 这个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可以分解为简单的动力学。 这种现象的物理意义在于它在扰动下的稳定性。 在过去，PI研究了方程的确定性扰动，目前的项目通过考虑存在（随机）噪声的稳定性，将这一主题引入了一个新的方向。 该项目的重点是在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点的几个问题。 PI的发现新的微观守恒定律KdV打开了大门，治疗三个看似无关的问题长期感兴趣的KdV线上：最佳正则性适定性，辛非压缩，和不变性的白色噪音。 此外，PI提出了一个连贯的计划，建立吉布斯测量的Landau-Lifshitz模型和白色噪声的不变性的聚焦立方非线性薛定谔方程（NLS）的不变性。 这个程序包括为与这些问题相关的物理原子模型建立类似的陈述（PI已经成功完成），然后为相应的粗略数据取连续极限。 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定性。 由于Landau-Lifshitz模型的Gibbs测度对应于球面上的布朗运动，因此从纯概率的角度来看，这个问题也很有趣，因为在这样的路径上产生了Hamilton测度保持流。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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

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

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