Wave Propagation and Scattering in Networks of Thin Waveguides

本计画的目的是分析波在薄波导网络中的传播。由于这种网络的三维几何结构复杂，这种类型的问题不能明确地解决，而且很难适用于计算方法。该项目的目标是开发一种方法，允许将问题简化为量子图上更简单的一维问题，这是波导直径趋于零时网络的极限。该研究将包括散射解的渐近分析，频谱，以及亥姆霍兹方程对所有频率（在底部和大部分频谱）的解析。该项目还包括分析光网络中波包减速的可能机制，以及研究系统在随机噪声存在下的稳定性和间歇性。该项目将为研究复杂光网络中的信息流提供数学背景。特别是，它将提供一种有效的工具来构建项链式设备，用于同步非常快的光学元件和非常慢的电子元件的工作。该项目的研究成果和方法可以应用于光子晶体、纳米技术和其他物理和技术领域的研究。

The objective of this project is the analysis of wave propagation in networks of thin waveguides. Problems of this type cannot be solved explicitly and are poorly amenable to computational methods because of the complicated 3-dimensional geometrical structure of such a network. The goal of the project is to develop methods allowing the reduction of the problem to a much simpler one-dimensional problem on the quantum graph, which is the limit of the network as the diameter of the waveguides tends to zero. The study will include the asymptotic analysis of the scattering solutions, the spectrum, and the resolvent for the Helmholtz equation for all frequencies (at the bottom and in the bulk of the spectrum). The project also includes the analysis of possible mechanisms for slowing down of wave packages in optical networks and the study of stability and intermittency in the system in the presence of random noise.The project will provide mathematical background for the investigation of information flows in complex optical networks. In particular, it will provide an efficient tool to construct necklace type devices for synchronizing the work of very fast optical elements and much slower electronics. The results and the methods to be developed in the project can be applied to the study of photonic crystals, nanotechnology, and other areas of physics and technology.