Stability of Solitary Waves in Dynamical Systems

动力系统中孤立波的稳定性

• 批准号: 1614734
• 负责人: Atanas Stefanov
• 金额: $ 19.3万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614734&HistoricalAwards=false
• 关键词: Stability; Solitary; Waves; Dynamical; Systems

Nonlinear wave equations give a mathematical description for many phenomena in optics, fluid dynamics, and a variety of other physical systems. This research is aimed at the mathematical study of special and very important "solitary-wave" solutions that represent a single wave or a train of single waves traveling through the medium. Solutions of this mathematical nature are instrumental for predicting behavior and designing engineering devices for the wide range of physically unrelated phenomena, such as water wave dynamics, in particular, gigantic ocean waves ("rogue" waves), magnetization of materials, propagation of light in optical fibers and other optical media, or some quantum mechanical dynamics. Even when it is not possible to completely determine the solutions to such equations, the evolution of the corresponding physical system can often be understood satisfactorily by considering a computational or approximate solution. This is feasible, provided the behavior of solutions does not change qualitatively when unavoidable computational errors or uncertainty in the parameters are introduced. In mathematics this insensitivity to small perturbations is termed "stability." The principal goal of this research project is developing new mathematical tools for study nonlinear dispersive waves and stability of their solitary-wave solutions. In particular, the Principal Investigator (PI) aims at obtaining precise, quantitative information about the long time (asymptotic) behavior. Graduate students will be trained and mentored through their participation in this project.The PI will consider the questions of stability of solitons, such as traveling waves, standing waves, traveling kinks, for various models arising in physical applications. Of particular concern will be the close-to-soliton behavior of the Dirac equations, the Ostrovsky and the short pulse equations, as well as various water wave equations. In addition, the PI will address several outstanding problems in the theory of spatially discrete dispersive systems, of the type of the discrete nonlinear Schroedinger equation and the granular chain model with Hertzian interactions. These are the models, where new paradigms are expected to emerge. More precisely, the models present obstacles to their investigation that are either not present in the corresponding "standard" continuous limits or else, the solutions behave in a substantially different ways than the respective continuous analogues. Thus, new mathematical methods need to be developed to address the challenges associated with the study of evolution of these discrete models.

非线性波方程对光学，流体动力学和各种其他物理系统的许多现象给出了数学描述。这项研究针对特殊且非常重要的“孤立波”解决方案的数学研究，该解决方案代表了单波浪或一系列通过介质传播的单波。这种数学性质的解决方案有助于预测行为和设计工程设备，以针对广泛的物理无关现象，例如水波动力学，尤其是巨大的海浪（“流氓”波（“流氓”波），材料的磁化，光纤维中光的传播，光纤中光纤维的传播以及其他光学介质和其他光学介质和其他量子机械动力学。即使无法完全确定该方程的解决方案，通常也可以通过考虑计算或近似解决方案来令人满意地理解相应物理系统的演变。这是可行的，前提是当引入参数中不可避免的计算错误或不确定性时，解决方案的行为不会在质量上发生变化。在数学中，这种对小扰动的不敏感被称为“稳定性”。该研究项目的主要目标是开发新的数学工具，用于研究非线性分散浪潮及其单独波解决方案的稳定性。特别是，首席研究员（PI）旨在获得有关长时间（渐近）行为的精确定量信息。研究生将通过参与该项目的参与而受到培训和指导。PI将考虑唯一的稳定性问题，例如行进浪，站立浪，旅行扭结，用于在物理应用中引起的各种模型。特别令人担忧的是，狄拉克方程，奥斯特罗夫斯基和短脉冲方程以及各种水浪方程的近距离行为。 此外，PI将在空间离散的分散系统，离散的非线性Schroedinger方程的类型以及与Hertzian相互作用的颗粒状链模型中解决几个出色的问题。这些是预计新范式出现的模型。更确切地说，模型存在于其研究的障碍，它们要么在相应的“标准”连续限制中不存在，否则，解决方案的行为与各自的连续类似物的行为大不相同。因此，需要开发新的数学方法，以应对与这些离散模型进化相关的挑战。