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，但在每个介质中传播的波往往会显示出相似的特征行为。 这包括快速振荡的发展，称为色散冲击波，长时间传播的脉冲称为孤子，以及孤立的高振幅扰动称为流氓波。 用于理解和描述这些现象的数学模型是相似的，无论所研究的介质是什么。 这是一个重大的挑战，以构建一个有凝聚力的数学框架，一般足以捕捉跨不同的模型的共同特征，但灵活性足以揭示在特定情况下的细节。 该项目将利用最新的分析、渐进和数值方法，开发包括冲击波、孤立子和异常波在内的色散现象的增强模型，并应用于流体力学、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不可或缺的一部分。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，振荡冲击，这是足够强大的，以处理两个冲击衰减和那些分裂成列车的孤子在KdV方程和相关的模型。 (2)开发计算上可行的方法，用于计算，误差估计，全球现象，如NLS方程和相关模型中的多个分散冲击波的相互作用。 (3)建立适用于NLS、sine-Gordon和Ablowitz-Ladik方程等的高振幅异常波的新的通用理论。 这项工作将以新的方式扩展和联合收割机的最新方法，从微分方程，渐近，散射理论，复分析，潜在的理论，和数值分析。这一奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.