Impact of Perturbations on Ultra-Short Solitary Waves in Optical Media

该奖项支持的这项研究的目标是为研究光学介质中的超短孤立波的下一代数学和计算工具做出贡献。特别是，这项研究将有助于表征扰动的影响。为此，获奖者将研究三个模型：经典的带高阶项的三次非线性薛定谔方程和最近发展起来的两个具有孤立波解的非线性介质中的超短脉冲模型，即Sch？afer和Wayne在2004年推导的短脉冲方程(SPE)和Chung和Sch？afer在2007年推导的非局部短脉冲方程(NSPE)。在项目的第一部分，将研究孤立波相对于初始条件摄动的稳定性。第二部分将集中于将这些模型扩展到更复杂的线性和非线性响应函数。该项目的第三部分将致力于描述孤子--S对介质随机变化的响应。作为这项工作的一部分，将发展多时间尺度系统中粗粒度噪声的方法。这三个部分都需要分析和数值技术的结合。将开发一个基于C++的计算库来实施新方法，并将在互联网上免费提供。更广泛地说，这项研究不仅对光学重要，而且对非线性、非局域性和随机性相遇的各种科学领域都很重要。作为该项目的一部分，本科生将参与这项研究，并将为新的光通信数学课程开发课程材料。近年来，在创造和探测超快光脉冲方面的实验成功为一系列新的光学现象打开了大门，这些现象发生在非常小的尺度上，因此非常快。当前的光学技术允许设计其结构比标准光纤更复杂的光学设备。这些新器件展示了在标准光纤中从未见过的显著现象。高比特率电信、激光手术和超宽带发电将受益于这些进步。这些潜在的新应用产生了对新的数学模型的需求，以在各种情况下正确地描述这种超快现象。这项研究的焦点是这样的光脉冲在通过非理想波导时是否保持稳定的问题。这项研究将使用获奖者及其合作者在2004年开发的一类数学模型。

The goal of the research that will be supported by this award is to contribute to the next generation of mathematical and computational tools for studying ultrashort solitary waves in optical media. In particular, this research will help characterize the impact of perturbations. For this purpose, the awardee will study three models: the classic cubic nonlinear Schr¨odinger equation with higher order terms and two recently developed models of ultra-short pulses in nonlinear media that both possess solitary wave solutions, namely the short-pulse equation (SPE) derived by Sch¨afer and Wayne in 2004 and the nonlocal short-pulse equation (NSPE) derived by Chung and Sch¨afer in 2007. In the first part of the project, the stability of the solitary waves with respect to perturbations of the initial conditions will be studied. The second part will focus on extensions of these models to more complicated linear and nonlinear response functions. The third part of the project will be devoted to the characterization of the soliton?s response to stochastic variations of the media. As a part of this work, methods to coarse-grain noise in systems with multiple time scales will be developed. All three parts will require a combination of analytical and numerical techniques. A C++ based computational library will be developed to implement the new methods and will be made available freely on the Internet. More broadly, the research will be important not just for optics, but for a variety of scientific areas in which nonlinearity, nonlocality, and randomness meet. As part of the project, undergraduate students will participate in the research, and course material for a new class on the mathematics of optical communications will be developed. In recent years, experimental success in the creation and detection of ultra-fast optical pulses has opened the door to a new range of optical phenomena that take place on very small scales and hence are extremely fast. Current optical technology allows to design optical devices whose structures are more complex than standard optical fibers. These new devices exhibit remarkable phenomena never seen in standard optical fibers. High bit-rate telecommunications, laser surgery and ultra-broadband generation will benefit from these advances. These potential new applications generate the need for novel mathematical models that describe the such ultra-fast phenomena correctly in a variety of situations. The focus of this research is the question whether such light pulses will remain stable as they propagate through non-perfect wave guides. The research will use a class of mathematical models that was developed by the awardee and his collaborators in 2004.