The Korteweg-de Vries Equation and Beyond

柯德维格-德弗里斯方程最初是在19世纪90年代引入的，用来简单有效地解释在浅水航道表面观测到的孤立波。孤立波在保持其原始形状的同时，可以传播很远的距离。这是一个例外，代表了两种现象之间的微妙相互作用：(1)扩散，导致更长的海浪传播得更快(例如，海啸的传播速度比海滩上看到的风浪快得多)。(2)非线性，它使大振幅波的传播速度快于短波(这一效应也表现在波在更深的水中传播得更快)。这两种效应在许多波系统中都很重要，控制它们之间的相互作用在光纤通信等技术中也扮演着重要的角色。Korteweg-de Vries方程及其密切关系提供了一个简单而优雅的框架，既可以加深我们对这些效应的理解，也可以教育处于各个准备阶段的未来科学家关于这些重要主题的知识。色散PDE的不变测度和随机强迫的研究在色散PDE的低正则性理论中提出了具有挑战性的问题。这个项目的一个主要主题是利用M.Visan和PI最近开发的新技术来解决Korteweg-de Vries方程(KdV)中的这些问题。在它们目前的形式中，这些方法以一种重要的方式利用了KdV的完全可积性。选择要研究的问题级数的一个关键目标是调查这些方法对破坏这种完全可积性的扰动有多健壮。该项目的第二个主要部分是将新方法应用于其他完全可积系统，最著名的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对门槛解决方案的研究，即那些位于分散和更复杂行为之间的解决方案。该奖项反映了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。

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.