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

扰动对光介质中超短孤立波的影响

• 批准号: 0807396
• 负责人: Tobias Schaefer
• 金额: $ 9.2万
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807396&HistoricalAwards=false
• 关键词: Impact; Perturbations; Ultra; Short; Solitary

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.

该奖项支持的研究目标是为研究光学介质中的超短孤立波提供下一代数学和计算工具，特别是，这项研究将有助于表征扰动的影响。获奖者将研究三个模型：具有高阶项的经典三次非线性薛定谔方程和两个最近开发的非线性介质中超短脉冲模型，它们都具有孤立波解，即由下式导出的短脉冲方程（SPE）： Schâfer 和 Wayne 在 2004 年提出了非局域短脉冲方程 (NSPE)，由 Chung 和 Schâfer 在 2007 年推导。在该项目的第一部分中，孤立波相对于初始条件扰动的稳定性将第二部分将重点研究这些模型对更复杂的线性和非线性响应函数的扩展，该项目的第三部分将致力于孤子响应的表征。作为这项工作的一部分，将开发具有多个时间尺度的系统中的粗粒度噪声的方法，这三个部分将需要开发基于 C++ 的计算库。更广泛地说，这项研究不仅对光学很重要，而且对非线性、非局域性和随机性相结合的各种科学领域也很重要。 ，本科生将参加近年来，超快光脉冲的产生和检测的实验成功为发生的一系列新的光学现象打开了大门。当前的光学技术允许设计比标准光纤更复杂的光学设备，这些新设备表现出在标准光纤中从未见过的显着现象。超宽带一代将受益于这些进步。潜在的新应用需要新的数学模型来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些光脉冲在通过非完美波导传播时是否会保持稳定。该研究将使用获奖者及其合作者于 2004 年开发的一类数学模型。