Universality in Nonlinear Waves and Related Topics

本项目在几种物理环境中的大波运动模型的背景下研究普遍性现象，例如地表水波和光纤中的电磁波。普遍性指的是，尽管波是由非常不同的机制，甚至是在非常不同的物理系统中启动的，但出现相同或非常相似的波型的情况。例如，海面上的无赖海浪经常被描述为由一个巨大的中心峰组成，两边都有一个明显的倾角。无赖波出现的条件并不重要--模式几乎总是相同的。这项研究的目的是确定应该预期的模式，因为这些知识可以用于从设备设计到减灾的各种应用，以及它们发生的原因。此外，由于建模是一个涉及大量特殊假设的过程，因此重要的是要了解模型预测的哪些功能独立于这些假设，并且通用性触及了这个问题的核心。研究生和职业生涯早期的研究人员将加入这项研究，这项研究将增强其影响，超越科学探索，进入下一代科学家的教育和培训。从数学上讲，普适性的研究与渐近分析有关，特别是当模型或解中的参数变大时，涉及双比例限制以定位关注点附近的坐标。研究人员将在由可积发展方程给出的非线性波的各种渐近模型中研究这种双标度极限。将范围略微扩大，将讨论涉及非线性波数学模型的渐近分析的几个具体问题，包括：(I)确定散焦非线性薛定谔方程和中间长波方程和Benjamin-Ono方程解的小色散渐近性；(Ii)分析聚焦非线性薛定谔方程的一族新的超越解的特征，称为“无穷级无赖波”；(Iii)研究Painlevé方程特解族的退化。研究人员将结合并进一步发展可积系统和渐近分析领域的技术来解决这些问题。预期的结果包括：首次证明了散焦的非线性薛定谔方程中的普遍波破裂，关于中间长波方程(在浅水Korteweg-de Vries极限和深水Benjamin-Ono极限之间内插的分层流体中的内波的非局部模型)解的小弥散渐近性态的新结果，发展了用于非局部Riemann-Hilbert问题的渐近分析的新的分析方法，以及关于Painlevé方程解空间和聚焦非线性薛定谔方程的新信息的发现。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。

This project studies the phenomenon of universality in the context of models for the motion of large waves in several physical settings such as surface water waves and electromagnetic waves in optical fibers. Universality refers to situations in which the same or very similar wave patterns appear despite the waves being set into motion by quite different mechanisms, or even in quite different physical systems. For instance, rogue waves on the sea surface are frequently characterized as consisting of a large central peak with a distinctive dip on either side. It does not matter much the conditions under which the rogue wave appears — the pattern is nearly always the same. The aim of this research is to determine what patterns should be expected, as this knowledge can be used in applications ranging from device design to disaster mitigation, and why they occur. Furthermore, because modeling is a process that involves numerous ad-hoc assumptions, it is important to understand which features predicted by a model are independent of those assumptions, and universality gets to the heart of this question. Graduate students and early-career researchers will join the investigator in this study, which enhances its impact beyond scientific inquiry and into education and training of the next generation of scientists. Mathematically, the study of universality is related to asymptotic analysis, specifically involving double-scaling limits to localize the coordinates near a point of interest, while a parameter in the model or solution becomes large. The investigator will study such double-scaling limits in various asymptotic models for nonlinear waves given by integrable evolution equations. Broadening the scope slightly, several specific questions involving asymptotic analysis of mathematical models for nonlinear waves will be addressed, including (i) determining the small-dispersion asymptotics of solutions of the defocusing nonlinear Schrödinger equation and the intermediate long-wave and Benjamin-Ono equations; (ii) analyzing the features of a new family of transcendental solutions of the focusing nonlinear Schrödinger equation termed "rogue waves of infinite order"; (iii) studying the degeneration of specific solution families of Painlevé equations. The investigator will combine and further develop techniques from the fields of integrable systems and asymptotic analysis to address these questions. Anticipated outcomes include a first proof of universal wave breaking in the defocusing nonlinear Schrödinger equation, new results on the small-dispersion asymptotic behavior of solutions of the intermediate long-wave equation (a nonlocal model for internal waves in stratified fluids interpolating between the shallow-water Korteweg-de Vries limit and the deep-water Benjamin-Ono limit), development of a new analytical technique for asymptotic analysis of nonlocal Riemann-Hilbert problems, and the discovery of new information about the solution space of Painlevé equations and the focusing nonlinear Schrödinger equation.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.