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，称为线性叠加定律，其中多个波形加在一起，但实际上许多波具有大振幅并且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，研究人员将使用物理实验和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。这种跨学科的研究将提供非线性波相互作用的基本理解与应用到流体动力学，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。研究人员将数学和实验研究非线性波的相互作用。 这将通过构建精确的和数值的孤子椭圆余弦（周期性行波）呼吸器解决方案，解决广义黎曼问题碰撞椭圆余弦波，获得调制方程和多维和强迫非线性波的解决方案，并进行流体实验。 采用的方法包括Whitham平均，可积系统方法（达布变换），守恒律方法，多尺度渐近，数值计算，并在粘性核心环状流的实验。 为了描述孤子和椭圆余弦波的复杂相互作用，该方法结合了双曲守恒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，范围从完全可积的，例如，Korteweg-de弗里斯和Kadomtsev-Petviashvili，到不可积，例如，Ostrovsky，2D Benjamin-Ono和导管方程。 建议的粘性核-环实验将提供定量测试的建模和理论预测。 发展的理论将有助于非线性波领域和大气和海洋内部和表面波propagation.This奖项的理解反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.