Integrable and Non-Integrable Dispersive Partial Differential Equations

PI建议研究的模型之一是Korteweg-de Vries(KdV)方程。这个方程式是一百多年前推导出来的，用来解释浅水航道中长波的行为。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，这个方程具有丰富的新奇特征，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，但在最小假设下解的存在直到最近才被PI和她的合作者证明。他们工作中的一个因素是最近发现了新的守恒定律。这个项目概述了现在可以使用这一发现攻击的其他几个问题。这个项目背后的另一个主要推动力是证明复杂的瞬时动力学在遥远的未来可以分解为简单的动力学。这一现象的物理意义取决于它在微扰下的稳定性。过去，PI研究了方程的确定性扰动，而当前的项目通过考虑(随机)噪声存在时的稳定性，将这一主题带到了一个新的方向。该项目主要研究位于非线性色散偏微分方程组、完全可积系统和随机偏微分方程交点的几个问题。PI发现了KdV的新的微观守恒律，为处理KdV的三个看似无关的问题打开了大门：最优正则性、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测度不变性和聚焦立方非线性薛定谔方程(NLS)的白噪声不变性的相干计划。该程序包括为与这些问题相关的物理原子模型建立类似的语句(PI已经成功地完成)，然后对相应的粗略数据取连续统极限。这应该会揭示Landau-Lifshitz模型和立方NLS模型的物理重整化，这些模型将确保此类数据的适定性。由于Landau-Lifshitz模型的Gibbs度量对应于球面上的布朗运动，从纯粹概率的角度来看，这个问题也很有趣，因为在这样的路径上产生了哈密顿度量保持流。这个奖项反映了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。

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.