Nonlinear Wave Motion

非线性波动

• 批准号: 9703850
• 负责人: Mark Ablowitz
• 金额: $ 11.1万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703850&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

9703850 Ablowitz Research involving a class of physically significant nonlinear systems arising in the study of nonlinear wave motion is proposed. Included in the studies are a new class of localized multidimensional solutions which can be related to a topological number; i.e. an index or winding number. Associated with the self dual Yang-Mills equations is a novel class of nonlinear ordinary differential equations, referred to as Darboux-Halphen (DH) type systems. These equations can be integrated by associating them with, and analyzing, certain linear monodromy problems. The linear problems have monodromy data which are evolving. The solutions of DH systems can be extremely complicated, exhibiting natural boundaries and dense branching in the complex plane. In special cases the solutions are expressed in terms of automorphic functions. Frequently researchers use numerical simulation to obtain approximations to solutions of nonlinear wave equations. Research by the PI and collaborators has demonstrated that computational chaos can be induced from truncation and roundoff errors. Use of integrable systems is extremely important since analytical results can be compared with numerical simulation. The study of nonlinear wave equations has wide ranging applications in physics and engineering. Intimately related to the studies in this proposal is a class of equations, called integrable systems. The mathematics of integrable systems and their solutions have been shown to play an important and fundamental role in our understanding of a variety of real world phenomena. Integrable systems usually admit a special class of extremely stable, localized wave solutions, referred to as solitons, which arise frequently in physical contexts. For example, in the study of fluid dynamics, huge soliton internal ocean waves, sometimes a mile in length, and hundreds of feet in amplitude, have been located in many parts of the ocean; e.g. near the coast of Oregon. Such waves can be devastating to anything caught in their path, hence knowledge of where such internal solitons are located in the ocean is of interest. Another application of solitons is in the study and designing of efficient long distance communication systems. Solitons eliminate the need for electronic signal regenerators which can seriously inhibit data rates. Researchers have used solitons successfully in the laboratory and currently they are being used in prototype communication systems. It may well be that future high data rate communications systems will be designed to propagate multiple soliton waves.

9703850 Ablowitz提出了涉及非线性波动研究中产生的一类物理上重要的非线性系统的研究。这些研究包括一类新的局部多维解，它可以与拓扑数相关；即指数或圈数。与自对偶Yang-Mills方程相关的是一类新的非线性常微分方程，称为Darboux-Halphen （DH）型系统。通过将这些方程与某些线性单问题联系起来并加以分析，可以对它们进行积分。线性问题具有不断演化的单态数据。DH系统的解可能非常复杂，在复平面上表现出自然的边界和密集的分支。在特殊情况下，解用自同构函数表示。研究人员经常使用数值模拟来获得非线性波动方程的近似解。PI和合作者的研究表明，计算混沌可以由截断和舍入误差引起。使用可积系统是非常重要的，因为分析结果可以与数值模拟进行比较。非线性波动方程的研究在物理和工程中有着广泛的应用。与本提案的研究密切相关的是一类称为可积系统的方程。可积系统及其解的数学已经被证明在我们理解各种现实世界现象方面起着重要和基本的作用。可积系统通常承认一类特殊的极其稳定的局域波解，称为孤子，它们在物理环境中经常出现。例如，在流体动力学的研究中，在海洋的许多地方都发现了巨大的孤子内部海浪，有时长达一英里，振幅达数百英尺；在俄勒冈海岸附近。这种波对任何在其路径上被捕获的东西都是毁灭性的，因此了解这种内部孤子在海洋中的位置是很有意义的。孤子的另一个应用是研究和设计高效的远程通信系统。孤子消除了电子信号再生器的需要，电子信号再生器会严重抑制数据速率。研究人员已经在实验室中成功地使用了孤子，目前它们正在原型通信系统中使用。未来的高数据速率通信系统很可能被设计成传播多个孤子波。