Wave-breaking and Resonant Phenomena

在项目的第一部分，破波将在聚焦非线性薛定谔方程（NLS）的背景下进行研究，这是一个著名的振幅相关光束自聚焦模型。NLS波串可能在非线性焦散点处突然中断，并重新排列成更复杂的波结构。这些结构在概念上可以看作是非线性波干涉，它们可以经历连续的断裂；此外，它们表现出调制不稳定性。控制不稳定性和了解破浪机理是项目的重点。半经典极限的Riemann-Hilbert问题（RHP）的最陡下降法将用于剥离RHP对NLS线性化的波形的主要贡献。最陡下降法本身将为此目的加以改进。在我们的方法中也涉及到数值成分。该项目的第二部分解决了波在具有重复结构的介质中传播时的共振行为。在之前的工作中，通过一个精确描述二维传输异常的数学公式，观察到通过这种介质板的传输异常，并将其与共振现象联系起来。目前的项目将开发共振附近状态有效密度的公式，并在所讨论的共振和Fano在1935年开创性量子力学论文中引入的共振之间建立牢固的联系（在这两种情况下，在对称破缺扰动之后，嵌入连续频谱的实特征值作为共振移动到较低的复半平面）。该项目还将通过对积分最陡下降法的详细使用，解决线性材料脉冲响应的物理学问题。了解波在天然材料和人造材料中的传播对这些材料的利用至关重要。第一个项目（破波）应用于技术上重要的材料中的波传播，用聚焦非线性薛定谔方程很好地近似描述。这种材料是光纤，其中调制不稳定性对光纤通信起不利作用。利用目前的技术，所研究的断裂现象开始在纳米尺度上被观察到，并将在纳米线传播理论的发展中发挥作用。第二个项目（共振行为）是指光通过光子晶体的传播，光子晶体的重复结构可以引导、阻挡甚至捕获光。在一个实验中，光几乎正常地入射到光子晶体材料板上，所研究的透射异常包括随着频率的变化，透射系数急剧下降，随后急剧上升。这两个项目都将涉及到学生在数学分析，数值实验和物理实验之间的相互作用，弥合数学和物理社区中显著不同类型的直觉。

In the first part of the project, wave-breaking will be studied in the context of the focusing nonlinear Schroedinger equation (NLS), a well-known model of amplitude-dependent beam self-focusing. NLS wavetrains may come to a sudden break at nonlinear caustic points and rearrange themselves into more complex wave structures. These structures, which may be visualized conceptually as nonlinear wave interference, may go through successive breaking; moreover, they display modulational instability. Controlling the instability and understanding mechanisms of the wave-breaking are the focus of the project. The steepest descent method for Riemann-Hilbert problems (RHP) in the semiclassical limit will be used to peel off the dominant contribution to the waveform from the RHP that linearizes NLS. The steepest descent method will itself be refined for this purpose. A numerical component is also involved in our approach. The second part of the project addresses resonant behavior in wave propagation through media that have a repetitive structure. In previous work, anomalies in transmission through slabs of such media were observed and linked to resonant phenomena through a mathematical formula describing the transmission anomaly accurately in 2D. The current project will develop formulae for the effective density of states near resonance and make a firm link between the resonance in question and the resonance introduced by Fano in a seminal 1935 quantum mechanics paper (in both cases, following a symmetry breaking perturbation, a real eigenvalue imbedded in continuous spectrum moves into the lower complex half plane as a resonance). The project will also resolve a question in physics on the pulse response of linear materials by an elaborate use of the method of steepest descent for integrals.The understanding of the propagation of waves in natural and in man-made materials is crucial to the utilization of these materials. The first project (wave-breaking) applies to wave propagation in technologically significant materials, described to a good approximation by the focusing nonlinear Schroedinger equation. Such materials are optical fibers in which the modulational instability acts adversely to fiber-optical communication. With current technology, the breaking phenomena researched are beginning to be observable at the nanoscale and will play a role in the development of a theory of propagation in nanowires. The second project (resonant behavior) refers to the propagation of light through photonic crystals, whose repetitive structure may guide, block, or even trap light. The transmission anomalies studied consist of a sharp dip followed by sharp rise of the transmission coefficient as the frequency is varied in an experiment in which light is incident nearly normally onto a slab of photonic crystal material. Both projects will involve students in interplay between mathematical analysis, numerical experiments, and physical experiments bridging the significantly different types of intuition in the mathematics and physics communities.