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，负责人：Mark Ablowitz，加州大学 题目：非线性波动 通过解析法、渐近法和计算方法，研究应用中经常出现的一类非线性波动方程及相关非线性系统的解和性质。 将研究多维方程和相关线性散射问题的新解。典型系统是 Kadomtsev-Petviashvili (KP) 方程，它是 Korteweg-deVries 方程的二维一维扩展。与 KP 方程的线性化相关的是非平稳薛定谔方程，它本身就是数学和物理学中的一个重要方程。 PI 最近的重要发现包括找到 KP 方程的新实数、局部多集解以及非平稳薛定谔方程的新类特征函数。这些解与一个正整数有关，称为电荷，它是一种缠绕数或指数。这些解决方案在电荷和其他指标方面的表征将继续进行。我们将寻求新类别的 KP 解决方案。四维自对偶杨米尔斯 (SDYM) 系统被视为“主”可积系统，其约简引发了新的非线性常微分方程的研究，其解具有不寻常的特征。特殊情况是一般情况下的经典 Darboux-Halphen 系统和 Chazy 方程。这些系统的解与模/自同构函数相关；就 Chazy 而言，它与众所周知的拉马努金函数有关。涉及新的 SDYM 削减的研究将继续进行。对微分非线性薛定谔 (NLS) 方程的研究表明，先前导出的标量差分 NLS 方程的新矢量扩展具有解解，并且可通过逆散射变换进行积分。标量和矢量差分 NLS 系统在连续极限上减少到物理上重要的 NLS 方程。将研究该矢量差分 NLS 方程的新解和性质。最近对水波的实验和理论研究表明，周期波的调制表现出不可重复的混沌动力学，而局域解不具备这些特性。这项工作的动机是 PI 早期关于计算混沌的研究。目前的研究表明，这种现象也发生在非线性光学中，并且似乎具有普遍性。这种无限维且可能普遍的混沌动力学将被详细研究。大振幅波系统的动力学通常被称为非线性波动。与小振幅现象不同，小振幅现象有大量广泛的理论可用，非线性波动的数学研究仍处于发展的早期阶段。非线性波动方程，例如本提案中描述的方程，在许多物理应用中至关重要。两个例子是水波和非线性光学，包括光纤通信。 极其稳定的局域非线性波（称为孤子）是与本项目的研究密切相关的一个主题。近年来，非线性光学的研究主要集中在孤子等局域大幅度脉冲的研究上。这种脉冲有多种用途，例如光束的整形和控制。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取得的数学发现现在正处于商业应用的风口浪尖。预计所有新结果的发表将发表在著名期刊上。