Frontiers in Dispersive Wave Equations

大气、水体、光纤、等离子体和玻色-爱因斯坦凝聚体都是非线性色散介质的例子。这意味着，虽然这些介质看起来彼此非常不同，但在每种介质中传播的波往往表现出相似的特征行为。这包括被称为色散激波的快速振荡的发展，被称为孤子的持久传播脉冲，以及被称为异常波的孤立的高振幅扰动。不管研究的是哪种媒介，用于理解和描述这些现象的数学模型都是相似的。构建一个有凝聚力的数学框架是一个主要的挑战，它足够通用，可以捕捉不同模型之间的共同特征，但又足够灵活，可以在特定情况下揭示细节。利用最新的分析、渐近和数值方法，该项目将开发包括激波、孤子和异常波在内的色散现象的增强模型，并将其应用于流体力学、气象学和光通信等领域。研究生、本科生和博士后研究人员的培训和专业发展将是工作的各个方面的组成部分。该项目的三个主要目标如下。(1)推导出基本色散结构的数学描述，即振荡激波，它足够健壮，可以处理KdV方程和相关模型中衰减的激波和分解成孤子序列的激波。(2)开发计算上可行的方法，计算具有误差估计的全局现象，如NLS方程和相关模型中的多个色散激波相互作用。(3)建立适用于NLS、sin - gordon和Ablowitz-Ladik方程等的新的、普遍的高振幅异常波理论。这项工作将以新的方式扩展和结合微分方程、渐近学、散射理论、复分析、势理论和数值分析的最新方法。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 atmosphere, bodies of water, optical fibers, plasmas, and Bose-Einstein condensates are all examples of nonlinear dispersive media. This means that while these media look very different from each other, waves propagating in each of them tend to display similar, characteristic behaviors. This includes the development of rapid oscillations known as dispersive shock waves, long-lasting propagating pulses known as solitons, and isolated, high-amplitude disturbances known as rogue waves. The mathematical models used to understand and describe these phenomena are similar, regardless of the media being studied. It is a major challenge to construct a cohesive mathematical framework general enough to capture common features across different models yet flexible enough to reveal details in specific situations. Drawing on the most up-to-date analytical, asymptotic, and numerical methods, the project will develop enhanced models of dispersive phenomena including shock waves, solitons, and rogue waves, with applications to areas including hydrodynamics, meteorology, and optical communications. The training and professional development of graduate students, undergraduate students, and postdoctoral researchers will be integral to all aspects of the work.The three major objectives of the project are as follows. (1) Derive a mathematical description of the basic dispersive structure, the oscillatory shock, that is robust enough to handle both shocks that decay and those that break up into trains of solitons in the KdV equation and related models. (2) Develop computationally feasible methods for computing, with error estimates, global phenomena such as the interaction of multiple dispersive shock waves in the NLS equation and related models. (3) Establish a new, universal theory of high-amplitude rogue waves applicable to the NLS, sine-Gordon, and Ablowitz-Ladik equations, among others. The work will extend and combine in new ways the latest methods from differential equations, asymptotics, scattering theory, complex analysis, potential theory, and numerical analysis.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.