Nonlinear Waves in Uniform and Periodic Media

这个项目的主要目标有三个方向。(A)改进现有的方法和发展新的方法，以求可积非线性波动方程，主要是非线性薛定谔方程，在小的色散极限下的严格渐近解；也了解解的结构，严重地受调制不稳定性的影响，如何依赖于初始数据的光滑度。(B)在可积和近可积波动方程理论范围内发展处理边界反射和随时间变化的强迫的方法。(C)开发数值和分析工具，用于研究周期和随机光子晶片中的电磁共振场和表面波现象，以及是否存在非线性。可积波传播通常是物理波在实际材料中传播的一个很好的近似(例如，非线性薛定谔方程及其微扰模型光在光纤中的传输)，而相关的数学方面，如调制不稳定性，在通信系统的优化设计中起着至关重要的作用。80年代首次开发的光子晶体，允许光被捕获或引导到与其波长相当的尺寸的区域。它们已经在微波天线、低阈值激光器和其他设备中使用。在它们目前的大多数应用中，它们都起着线性器件的作用。我们研究非线性行为，因此特别关注共振行为，它产生了大的场，其中介质的非线性起着核心作用。

The main aims of this project are in three directions. (a) To advance existing approaches and to develop new methods for the rigorous asymptotic solution of integrable nonlinear wave equations, primarily the nonlinear Schroedinger equation, in the small dispersion limit; also to understand how the structure of the solution, critically affected by the modulational instability, depends on the degree of smoothness of the initial data. (b) To develop methods within the theory of integrable and near-integrable wave equations to deal with wave reflection from boundaries and time dependent forcing. (c) To develop numerical and analytical tools for the study of electromagnetic resonant fields and surface wave phenomena in periodic and random photonic crystal slabs with as well as without the presence of nonlinearity. INTEGRABLE WAVE PROPAGATION is often a very good approximation to physical wave propagation in real materials (e.g. the nonlinear Schroedinger equation and its perturbations model transmission of light in fiber optics), and related mathematical aspects like the modulational instability play a crucial role in the optimal design of communications systems. PHOTONIC CRYSTALS, first developed in the eighties, allow light to be trapped in or guided through regions of dimensions comparable to its wavelength. They are already in use in microwave antennas, low threshold lasers and other devices. In most of their current applications they function as linear devices. We investigate nonlinear behavior and therefore pay particular attention to resonant behavior, which generates large fields for which the nonlinearity of the medium plays a central role.