A Study of Wave Patterns

大家都很熟悉，波浪通常会将自己组织成某些可识别的模式。例如，落入池塘的一块石头会产生同心的涟漪环，而船的尾迹则具有特有的V形。虽然它们在日常生活中可能更难看到，但其他类型的波，如电磁波，也经常产生这样的模式。这意味着，可以通过对几种不同类型的物理系统中常见的某些基本波动模型的数学研究来理解波动模式。以类似的方式，从第一性原理理解波型也是一个值得追求的问题，因为这种波型经常出现在许多物理环境中，从非常小的(例如原子尺度的光学或量子波)到非常大的(例如星系尺度的引力波)。这个项目是对波传播数学模型中出现的波型的系统调查，它的目的是对给定模型中可能出现的各种波型进行分类，并确定可能出现给定波型的模型范围和物理情况。作为一个实例应用，该项目的成果将有助于我们理解无赖海浪，即突然出现又突然消失的海面大扰动，但除非预料到并避免或以其他方式减轻，否则可能对船舶造成巨大损害。该项目的一些部分将作为培训初级研究人员的工具，如本科生、研究生和博士后。该项目旨在发展和应用从完全可积系统理论到波型研究的方法。关键的数学方法是复数、泛函和渐近分析的工具，研究的结果将与非线性波传播、可积系统和特殊函数领域相关。要研究的具体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方程组。该项目将研究规则波型，即由平滑初始条件自发产生的调制波列，以及普遍波型，即出现在不依赖于初始条件(甚至可能不依赖于运动方程)的双尺度极限中的特殊结构。具体目标包括开发通过渐近分析在由多个耦合场或多个空间维度组成的可积系统中调查波形的新技术。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。

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.