Nonlinear Wave Motion

非线性波动

• 批准号: 0303756
• 负责人: Mark Ablowitz
• 金额: $ 21.27万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0303756&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

Abstract: 0303756, PI: Mark Ablowitz, University of CaloradoTitle: Nonlinear Wave MotionThe solutions and properties of a class of nonlinear wave equations and related nonlinear systems which arise frequently in application will be studied by analytical, asymptotic and computational methods. New solutions of multi-dimensional equations and related linear scattering problems will be investigated. A prototypical system is the Kadomtsev-Petviashvili (KP) equation, which is a two-space one-time dimensional extension of the Korteweg-deVries equation. Associated with the linearization of the KP equation is the nonstationary Schrodinger equation which itself is a prominent equation in mathematics and physics. Important recent discoveries by the PI include finding new real, localized, multi-lump solutions to the KP equation and new classes of eigenfuctions to the nonstationary Schrodinger equation. These solutions are related to a positive integer, referred to as the charge, which is a type of winding number or index. The characterization of these solutions in terms of the charge and other indices will continue. New classes of KP solutions will be sought. Reductions of the four dimensional self-dual Yang Mills (SDYM) system, which is viewed as a "master" integrable system, leads to the study of novel nonlinear ordinary differential equations whose solutions possess unusual features. Special cases are the classical Darboux-Halphen system and Chazy equation, in general position. The solutions of these systems are related to modular/automorphic functions; and in the case of Chazy, it is related to the well known Ramanujan functions. Research involving new reductions of SDYM will continue. The investigation of differential-difference nonlinear Schrodinger (NLS) equations has shown that new vector extensions of a previously derived scalar difference NLS equation has soltion solutions and is integrable by the inverse scattering transform. The scalar and vector difference NLS systems reduce in the continuous limit to the physically important NLS equations. New solutions and properties of this vector difference NLS equation will be studied. Recent experimental and theoretical studies of water waves has shown that modulation of periodic waves exhibit nonrepeatible, chaotic dynamics whereas localized soltion soltuions do not possess these properties. This work was motivated by earlier research by the PI on computational chaos. Current research indicates that this phenomena also occurs in nonlinear optics and appears to be universal in character. This infinite dimensional and possibly universal chaotic dynamics will be studied in detail.The dynamics of wave systems with large amplitude is often referred to as nonlinear wave motion. Unlike small amplitude phenomena where substantial and wide ranging theory is available, the mathematical investigation of nonlinear wave motion is still at an early stage of development. Nonlinear wave equations, such as the ones described in this proposal, are centrally important in many physical applications. Two examples are water waves and nonlinear optics, including fiber optic communications. Extremely stable, localized nonlinear waves called solitons, is a subject which is closely related to the research investigations in this project. The study of nonlinear optics has focused in recent years on the study of localized large amplitude pulses such as solitons. Such pulses, are used in a variety of ways such as the shaping and controlling of light beams. In fiber optic communications, understanding the properties of large amplitude optical pulses are important for the next generation of communication systems. The mathematical discoveries made in the field of nonlinear fiber optic waves only a few years years ago are now at the cusp of commercial application. It is expected that publication of all new results will be published in prominent journals.

摘要：0303756，PI：Mark Ablowitz，University of Calorado题目：非线性波动本文将用解析、渐近和计算的方法研究应用中经常出现的一类非线性波动方程及其相关非线性系统的解及其性质。 多维方程和相关的线性散射问题的新的解决方案将被调查。Kadomtsev-Petviashvili（KP）方程是Korteweg-deVries方程的两空间一维扩展。与KP方程线性化相联系的是非定常薛定谔方程，它本身就是数学和物理中的一个重要方程。PI最近的重要发现包括找到KP方程的新的真实的、局部化的多块解和非平稳薛定谔方程的新的本征函数类。这些解决方案都涉及到一个正整数，称为电荷，这是一种类型的缠绕数或指数。将继续根据电荷和其他指数对这些溶液进行表征。将寻求新类别的金伯利进程解决方案。四维自对偶杨米尔斯（SDYM）系统是一个“主”可积系统，通过对该系统的约化，可以研究一类新的非线性常微分方程，其解具有特殊的性质.特殊情况是经典的Darboux-Hézen系统和Chazy方程，在一般情况下。这些系统的解与模/自守函数有关;在Chazy的情况下，它与众所周知的Ramanujan函数有关。将继续进行涉及SDYM新减少量的研究。对微分-差分非线性薛定谔方程的研究表明，一个标量差分非线性薛定谔方程的新的矢量扩张具有解，并且通过逆散射变换是可积的。标量和矢量差分NLS系统在连续极限下约化为物理上重要的NLS方程。本文将研究这类向量差分NLS方程的新解及其性质。最近的实验和理论研究的水波表明，调制的周期波表现出nonrepeatible，混沌动力学，而本地化solution soltuions不具备这些属性。这项工作的动机是由PI对计算混沌的早期研究。目前的研究表明，这种现象也发生在非线性光学，似乎是普遍的性质。这种无限维的、可能是普遍的混沌动力学将被详细地研究。具有大振幅的波动系统的动力学通常被称为非线性波动。与小振幅现象不同，非线性波动的数学研究仍处于早期发展阶段。非线性波动方程，如本建议中所描述的，在许多物理应用中非常重要。两个例子是水波和非线性光学，包括光纤通信。 被称为孤子的极其稳定的局域非线性波是与本项目的研究调查密切相关的课题。近年来，非线性光学的研究主要集中在孤子等局域大振幅脉冲的研究上。这样的脉冲以各种方式使用，例如光束的成形和控制。在光纤通信中，了解大幅度光脉冲的特性对于下一代通信系统非常重要。 几年前在非线性光纤波领域的数学发现现在正处于商业应用的风口浪尖。预计所有新成果将在著名期刊上发表。