The Korteweg-de Vries Equation and Beyond

Korteweg-de Vries方程最初（在19世纪90年代）被引入，是为了对浅水通道表面孤立波的观测给出一个简单有效的解释。孤波在保持其原始形状的情况下传播很远的距离。这是一种特殊现象，它代表了两种现象之间微妙的相互作用：(1)色散，这导致更长的波浪传播得更快（例如，海啸传播得比海滩上看到的风浪快得多）。(2)非线性，使振幅较大的波比振幅较小的波传播速度快（这种效应也表现在水深较深的波传播速度较快）。这两种效应在许多波系统中都很重要，控制它们的相互作用在诸如光纤通信等技术中也起着重要作用。Korteweg- de Vries方程及其密切关系提供了一个简单优雅的框架，既可以加深我们对这些效应的理解，也可以教育未来的科学家，在所有准备阶段，关于这些重要主题。色散偏微分方程的不变测度和随机强迫的研究对色散偏微分方程的低正则性理论提出了挑战。该项目的一个主要主题是通过采用M. Visan和PI最近开发的新技术来解决Korteweg-de Vries方程（KdV）设置中的这些问题。在它们目前的形式中，这些方法在很大程度上利用了KdV的完全可积性。选择待研究问题的级数的一个关键目标是研究这些方法对破坏这种完全可积性的扰动的鲁棒性。该项目的第二个主要组成部分是将新方法应用于其他完全可积系统，特别是非线性薛定谔方程（NLS）和修正的KdV方程（mKdV）。该项目的第三个方面是研究阈值解，即那些位于散射和更复杂行为之间的边界的解。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。

The Korteweg-de Vries equation was originally introduced (in the 1890s) to give a simple effective explanation of the observation of solitary waves on the surface of shallow channels of water. Solitary waves travel large distances while maintaining their original shape. This is exceptional and represents a subtle interaction between two phenomena: (1) Dispersion, which results in longer waves traveling faster (e.g. tsunami travel much faster than the wind waves seen at the beach). (2) Nonlinearity, which makes large amplitude waves travel faster than shorter waves (this effect also manifests in waves traveling faster in deeper water). These two effects are important in many wave systems and controlling their interactions plays an important role also in technology, such as fibre-optic communication. The Korteweg--de Vries equation and its close relations provide a simple elegant framework both for deepening our understanding of these effects and also for educating future scientists, at all levels of preparation, about these important topics.The study of both invariant measures for dispersive PDE and stochastic forcing raise challenging questions in the low-regularity theory of dispersive PDE. A major theme of this project is to tackle such problems in the setting of the Korteweg-de Vries equation (KdV) by employing new techniques developed recently by M. Visan and the PI. In their current form, these methods exploit the complete integrability of KdV in a significant way. A key goal in selecting the progression of problems to be studied is to investigate how robust these methods are against perturbations that destroy this complete integrability. A second major component of the project is the adaptation of the new methods to other completely integrable systems, most notably the nonlinear Schrodinger equation (NLS) and the modified KdV equation (mKdV). The third facet of the project is the study of threshold solutions, namely, those that lie at the boundary between scattering and more complicated behaviour.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.