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

该奖项支持的研究目标是为研究光学介质中超短孤波的下一代数学和计算工具做出贡献。特别是，这项研究将有助于描述扰动的影响。为此，得奖者将研究三个模型，包括经典的三次非线性薛定谔方程（含高阶项），以及两个新近发展的非线性介质中的超短脉冲模型，即Schüafer和韦恩于2004年推导的短脉冲方程（SPE）和Chung和Schüafer于2007年推导的非局域短脉冲方程（NSPE）。在该项目的第一部分，孤立波的稳定性相对于扰动的初始条件进行了研究。第二部分将集中在这些模型的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孤子的特性？的反应随机变化的媒体。作为这项工作的一部分，将开发多时间尺度系统中粗粒度噪声的方法。所有这三个部分都需要结合分析和数字技术。将开发一个以C++为基础的计算图书馆，以实施新的方法，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很重要，而且对非线性、非定域性和随机性相遇的各种科学领域也很重要。作为该项目的一部分，本科生将参与研究，并将开发一个新的光通信数学课程的教材。近年来，超快光脉冲的产生和探测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器件。这些新器件表现出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，这些模型可以在各种情况下正确描述这种超快现象。这项研究的重点是这样的光脉冲是否会保持稳定，因为他们通过非完美的波导传播的问题。这项研究将使用获奖者和他的合作者在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.