Wave-breaking and Resonant Phenomena

Riemann-Hilbert问题的最速下降法(积分计算的经典最速下降法的一种非线性模拟)是在渐近区域(如小离散或长时间)求解可积系统的选择方法。通过最陡下降法的新发展，研究了半经典聚焦非线性薛定谔方程(NLS)解的二次及更高阶破缺问题。该方法将允许复平面中的极点串(它们是Zakharov-Shabat特征值问题的解)以类似的方式变形，并用于类似于连续轮廓变形的目的。与我们以前的方法相反，两极的离散性将被保留，这种方法将两极浓缩到它们的连续极限，获得了第一次突破，但在处理第二次突破时会遇到严重的障碍。我们还将处理空间半直线上NLS波的推导，这是由边界点上的时间周期驱动器产生的。这里长期存在的挑战是，逆散射法是求解发展方程的自然方法。对于边值问题，该方法一方面需要超定的数据。我们建议通过由多相波和孤子(呼吸)构成的边界层将波列与驱动器匹配来克服这一困难。这与问题的线性极限是一致的。在提案的第二部分，我们继续开发关于光通过共振附近的光子晶体的传输和引导的代码开发和分析模型开发(见下文)。在提案的两个部分，我们讨论了天然和人造材料中的波，特别是这些波的产生、传播、破裂和在共振条件下的特殊行为。分析的物理和技术对应主要是光学(线性和非线性)。我们分析的一个中心对象是聚焦非线性薛定谔方程(NLS)，它主要出现在非线性光学传输中，以及它的许多变体。NLS是出了名的不稳定，表现出波列的自发分解和新波列的形成。虽然我们从以前的工作中了解了第一次破裂的机制，但随后的破裂和最终出现的波列的类型是具有挑战性的悬而未决的问题。这些问题构成了我们提案的两个焦点之一。提案的第二个重点涉及电磁波，特别是光及其与被称为光子晶体(PC)的人造材料的相互作用。这些金属或非金属的小规模材料具有周期性或重复的几何形状(例如，在周期性晶格的节点处穿孔的膜或板)。入射到PC上的单色光必须调整到每个PC组成材料内部的折射率。当PC几何结构适当时，光可以被操纵，以采取新颖和技术上可利用的模式进行响应。例如，光的引导、阻挡甚至捕获，以及通过PC薄膜的非凡传输。第一批涉及二维周期光子晶体的商业产品已经以光子晶体光纤(PCF)的形式出现。由于其能够将光限制在中空纤芯中或具有传统光纤无法实现的限制特性，PCF目前在光纤通信、光纤激光器、非线性器件、高功率传输、高灵敏度气体传感器等领域得到了应用。我们的努力是双重的。我们将继续开发专门用于PC结构的数值代码，并适合于克服共振附近的计算困难。我们已经验证了这类编码的性能明显优于商用EM编码。我们将实施这些代码来优化穿孔薄膜中的传输，并试验新的PC几何结构。我们还将继续开发简化的数学模型，以解析地捕捉当PC在非线性区域运行时现象的本质。

The steepest descent method for Riemann-Hilbert problems (a nonlinear analog of the classical steepest descent method for the evaluation of integrals) is the method of choice for the solution of integrable systems in asymptotic regimes, such as small dispersion or long time. We propose the study of the second and higher breaking of the solution of the semiclassical focusing nonlinear Schroedinger equation (NLS), through a new development of the steepest descent method. The method will allow the deformation of strings of poles in the complex plane (they are solutions of the Zakharov-Shabat eigenvalue problem) in similar ways and for similar purposes that continuous contours are deformed. The discreteness of the poles will be retained, contrary to our previous approach, that condensed the poles to their continuum limit, obtaining the first break, but leading to serious obstacles in treating the second break. We will also tackle the derivation of the NLS waves in a spatial half-line, that result from a time-periodic driver at the boundary point. The long-standing challenge here is that the method of inverse scattering method is natural for solving evolution equations. Confronted with a boundary value problem, the method demands overdetermined data on one side. We propose to overcome the difficulty by matching the wavetrain to the driver, through a boundary layer constructed of multiphase waves and solitons (breathers). This is consistent with the linear limit of the problem. In the second part of the proposal, we continue our code development and analytic model development on the transmission and guidance of light through photonic crystals near resonance (see below). In both parts of the proposal, we address waves in natural and in man-made materials, in particular, the generation, propagation, breaking and extraordinary behaviour under resonant conditions of these waves.The physical and technological counterparts of the analysis are mainly in optics (linear and nonlinear). A central object of our analysis is the focusing nonlinear Schroedinger equation (NLS), which appears dominantly in nonlinear optical transmission, together with its many variants. NLS is notoriously unstable, exhibiting the spontaneous break-up of wave-trains and the formation of new ones. While the mechanism of the first break is understood from our previous work, subsequent breaks and the type of wave-trains that eventually emerge are challenging open questions. These questions constitute one of the two focal points of our proposal. The second focal point of the proposal addresses electromagnetic (EM) waves, in particular light and its interaction with manufactured materials, known as photonic crystals (PC). These small-scale materials, metallic or non-metallic, have a periodic or repetitive geometry (e.g. film or plate perforated at the nodes of a periodic lattice). Monochromatic light, incident upon the PC, must adjust to the refractive index inside each of the PC constituent materials. When the PC geometry is structured appropriately, light can be manipulated to respond taking novel and technologically exploitable modes. Examples are the guiding, blocking and even trapping of light, as well as extraordinary transmission through PC films. The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of photonic-crystal fibers (PCF). Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic communications, fiber lasers, nonlinear devices, high-power transmission, highly sensitive gas sensors, and other areas. Our effort is two-fold. We will continue our development of numerical codes that are dedicated to PC structures and are appropriate for overcoming the difficulties of calculation near resonance. We have verified that such codes significantly outperform commercial EM codes. We will implement these codes to optimize transmission in perforated films and to experiment with novel PC geometries. We will also continue our development of simplified mathematical models that capture analytically the essence of phenomena when a PC operates in the nonlinear regime.