Collaborative Research: Collapse, Rogue Waves, and their Applications: From Theory to Computation and Beyond

该项目旨在进一步理解由非线性常微分方程、偏微分方程和晶格微分方程建模的系统中的坍缩型现象和异常波。坍缩型现象在数学上用解来描述，当它们的一些属性在有限时间内变得无界时，解保持自相似。自相似是指在适当的空间、时间尺度和解振幅下保持形状。坍缩现象与光学中光束的聚焦和原子物质波有关。异常波具有特征长度或时间尺度和极端振幅；它们在流体力学、非线性光学、原子和等离子体物理等学科中很重要。利用动力系统和计算技术，本项目旨在重新制定基础模型，并通过将相关模式视为自相似解决方案，为研究崩塌现象和异常波提供统一的方法。该项目预计将提供关于机制的见解，并在一些典型的数学模型中简化坍塌现象的数学描述，这些模型以这些潜在的灾难性聚焦事件为特征，以及在连续体和空间离散系统中极端波的形成、预测和分析。该项目将为学生提供研究培训机会。该项目将探索最近导出的用于研究非线性Schrödinger方程中心色散波模型的自聚焦波的范式，并将寻求相关模型（如Korteweg-de Vries方程）的推广。这种坍缩波的稳定性分析有望揭示这种系统的光谱特性和潜在的不稳定性，它们与对称性的联系，它们在不同环境下（超临界、临界和亚临界）的动力学含义，以及它们在原始“非爆炸”框架中的重新解释。该项目的第二个重点将是从动力系统的角度研究异常波，包括通过时间周期解的极限来表征稳定性和高阶色散设置下的异常波。理论分析将得到数值模拟的证实，包括基于通缩的不动点技术和伪弧长延拓，以及最先进的基于轮廓积分的特征值求解器。还将寻求与进行实验室实验的试验组的合作。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 project aims to further the understanding of collapse-type phenomena and rogue waves in systems that are modeled by nonlinear ordinary, partial, and lattice differential equations. Collapse-type phenomena are mathematically described by solutions that remain self-similar as some of their attributes become unbounded in finite time. Self-similarity refers to preservation of shape when an appropriate scaling of space and time and solution amplitude is employed. Collapse phenomena are relevant to the focusing of light beams in optics and to atomic matter waves. Rogue waves have a characteristic length or time scale and extreme amplitudes; they are important in subjects such as hydrodynamics, nonlinear optics, and atomic and plasma physics. Using dynamical systems and computational techniques, this project aims to reformulate the underlying models and provide a unified approach to studying both collapse phenomena and rogue waves by treating the relevant patterns as self-similar solutions. The project is expected to provide insights on the mechanisms and reduced mathematical descriptions of collapse phenomena in some of the prototypical mathematical models that feature these potentially catastrophic focusing events, as well as on the formation, prediction, and analysis of extreme waves in both continuum and spatially discrete systems. The project will offer research training opportunities for students. The project will explore a recently derived normal form for the study of self-focusing waves of the central dispersive wave model of the nonlinear Schrödinger equation and will seek generalizations for related models (such as the Korteweg-de Vries equation). Stability analysis of such collapsing waves is expected to shed light on the spectral properties and potential instabilities of such systems, their connection to symmetries, their implications for the dynamics in different settings (supercritical, critical, and subcritical), and their reinterpretation in the original "non-exploding'' frame. A second focus of the project will be the study of rogue waves from a dynamical systems viewpoint, including characterization of stability via limits of time-periodic solutions and rogue waves in higher-order dispersion settings. The theoretical analysis will be corroborated by numerical simulations involving deflation-based fixed-point techniques and pseudo-arclength continuation, as well as state-of-the-art contour integral-based eigenvalue solvers. Collaboration with experimental groups performing laboratory experiments will also be sought.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.