Nonlinear Wave Interactions

波的相互作用在自然界、实验室和技术中无处不在。虽然波相互作用的许多数学和科学依赖于小振幅波的基本性质，即所谓的线性叠加定律，在该定律中，多个波形相加在一起，但实际上许多波是大振幅的，并不符合这种线性行为。非线性波的一些例子包括近岸环境中的表面和内部重力波，实验室光纤实验中的强激光，以及超冷原子气体实验中的量子力学物质波。从多维相互作用的非线性波到非线性周期波和孤立波的相互作用，研究人员将利用物理实验和一种被称为调制理论的近似技术来研究非线性波相互作用的数学和物理。这项跨学科的研究将提供对非线性波相互作用的基本理解，并将其应用于流体动力学、非线性光学和量子流体。该项目将为本科生和研究生提供研究培训机会。研究人员将对非线性波相互作用进行数学和实验研究。这将通过构造精确的和数值的孤子-椭圆形(周期行波)呼吸解，求解碰撞椭圆波的广义黎曼问题，获得多维和强迫非线性波的调制方程和解，以及进行流体实验来实现。所采用的方法包括Whitham平均法、可积系统方法(Darboux变换)、守恒律方法、多尺度渐近法、数值计算和粘性岩心-环空流动实验。为了描述孤子和椭圆波的复杂相互作用，该方法结合了双曲守恒定律和非线性色散波的技术。要分析的方程是具有地球物理和技术意义的模型，从完全可积的，例如Korteweg-de Vries和Kadomtsev-Petviashvili，到不可积的，例如Ostrovsky、2D Benjamin-Ono和管道方程。所提出的粘性芯环实验将为模型和理论预测提供定量检验。发展的理论将有助于非线性波领域以及对大气和海洋内波和面波传播的理解。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。

Wave interactions are ubiquitous in nature, the laboratory, and technology. While much of the mathematics and science of wave interactions relies upon a basic property of small amplitude waves, known as the law of linear superposition, in which multiple wave shapes add together, in practice many waves are of large amplitude and do not conform to this linear behavior. Some examples of nonlinear waves include surface and internal gravity waves in the near-shore environment, intense laser light in laboratory fiber optic experiments, and quantum mechanical matter waves in experiments with ultracold atomic gases. From multidimensional interacting nonlinear waves to the interaction of nonlinear periodic waves and solitary waves, the investigator will study the mathematics and physics of nonlinear wave interactions using physical experiments and an approximation technique known as modulation theory. This interdisciplinary research will provide a fundamental understanding of nonlinear wave interactions with applications to fluid dynamics, nonlinear optics, and quantum fluids. The project will provide research training opportunities for undergraduate and graduate students. The investigator will mathematically and experimentally study nonlinear wave interactions. This will be achieved by constructing exact and numerical soliton-cnoidal (periodic traveling wave) breather solutions, solving generalized Riemann problems for colliding cnoidal waves, obtaining modulation equations and solutions for multidimensional and forced nonlinear waves, and performing fluid experiments. Methods to be employed include Whitham averaging, integrable system methods (Darboux transformations), conservation law methods, multiscale asymptotics, numerical computation, and experiments in viscous core-annular flows. To describe the complex interactions of solitons and cnoidal waves, the approach unites techniques from hyperbolic conservation laws and nonlinear dispersive waves. The equations to be analyzed are models of geophysical and technological significance, ranging from completely integrable, e.g., Korteweg-de Vries and Kadomtsev-Petviashvili, to non-integrable, e.g., Ostrovsky, 2D Benjamin-Ono, and conduit equations. The proposed viscous core-annular experiments will provide quantitative tests of the modeling and theoretical predictions. The developed theory will contribute to the field of nonlinear waves and to the understanding of atmospheric and oceanic internal and surface wave propagation.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.