A Study of Wave Patterns

每个人都熟悉波浪通常会将自己组织成某些可识别的模式。例如，一块石头掉进池塘里会产生同心圆的涟漪，船只的尾迹呈典型的v形。虽然它们在日常生活中可能很难看到，但其他类型的波，如电磁波，也经常产生这种模式。这意味着，波浪的模式可以从波浪运动的某些基本模型的数学研究中理解，这些模型在几种不同类型的物理系统中是常见的。同样，从第一性原理理解波的模式是值得追求的，因为这种模式经常出现在许多物理环境中，从很小的（例如原子尺度上的光学或量子波）到很大的（例如星系尺度上的引力波）。该项目是对波浪传播数学模型中出现的波浪模式的系统研究，其目的是对给定模型中可能出现的各种波浪模式进行编目，并确定给定模式可能出现的模型和物理情况的范围。作为一个应用实例，这个项目的成果将有助于我们对异常浪的理解，异常浪是海面上突然出现并消失的巨大扰动，但它会对船舶造成巨大损害，除非预测并避免或以其他方式减轻。该项目的某些部分将作为培训初级研究人员（如本科生、研究生和博士后）的工具。本项目旨在发展和应用完全可积系统理论的方法来研究波浪型。关键的数学方法是复杂分析、泛函分析和渐近分析的工具，研究结果将涉及非线性波传播、可积系统和特殊函数等领域。具体研究的模型包括三波共振相互作用方程、聚焦非线性薛定谔方程、Davey-Stewartson方程和Painleve方程族。该项目将研究规则波型，即平滑初始条件下自发产生的调制波列，以及通用波型，即出现在双尺度极限下的特殊结构，不依赖于初始条件（甚至可能不依赖于运动方程）。具体目标包括通过在由多个耦合场或多个空间维度组成的可积系统中进行渐近分析来研究波形的新技术的发展。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。

It is familiar to everyone that waves typically organize themselves into certain identifiable patterns. For instance, a stone dropped into a pond produces concentric rings of ripples, and the wake of a ship has a characteristic V-shape. Although they may be harder to see in everyday experience, other types of waves like electromagnetic waves frequently also produce such patterns. This means that wave patterns might be understood from the mathematical study of certain underlying models of wave motion that are common to several different types of physical systems. In a similar way, understanding wave patterns from first principles is a worthwhile pursuit because such patterns occur frequently in so many physical circumstances, ranging from the very small (e.g. optics or quantum waves on the scale of atoms) to the very large (e.g. gravitational waves on the scale of the galaxy). This project is a systematic investigation of wave patterns arising in mathematical models of wave propagation, and it aims both to catalogue the various wave patterns that can occur in a given model and to determine the range of models and physical situations in which a given pattern can arise. As one example application, the outcomes of this project will contribute to our understanding of rogue waves, large disturbances of the sea surface that appear out of nowhere and disappear just as suddenly, but which can cause great damage to ships, unless they are anticipated and avoided or otherwise mitigated. Some parts of the project will serve as vehicles for the training of junior researchers such as undergraduate students, graduate students, and postdocs. This project aims to develop and apply methods from the theory of completely integrable systems to the study of wave patterns. The key mathematical methods are tools of complex, functional, and asymptotic analysis, and the results of the research will be relevant to the fields of nonlinear wave propagation, integrable systems, and special functions. Specific models to be studied include the three-wave resonant interaction equations, the focusing nonlinear Schroedinger equation, the Davey-Stewartson equations, and the family of Painleve equations. The project will study both regular wave patterns, i.e., modulated wave trains spontaneously generated from smooth initial conditions, and universal wave patterns, i.e., special structures appearing in double-scaling limits that do not depend on initial conditions (and perhaps not even on the equation of motion). Specific goals include the development of new techniques for the investigation of wave patterns via asymptotic analysis in integrable systems consisting of multiple coupled fields or in multiple space dimensions.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.