Nonlinear Wave Motion

摘要：0303756，PI：Mark Ablowitz，Calorado大学题目：非线性波动运动将用解析、渐近和计算的方法研究一类在应用中经常出现的非线性波动方程和相关的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。Kadomtsev-Petviashvili(KP)方程是Kadomtsev-Petviashvili(KP)方程的一个原型系统，它是Korteweg-DeVries方程的二维一维推广。与KP方程的线性化相关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方程的新的实数、局部化、多集总解和非定常薛定谔方程的新的本征函数类。这些解与称为电荷的正整数有关，该正整数是一种绕组数或指数。根据电荷和其他指数对这些解决方案的表征将继续下去。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)系统被视为一个“主”可积系统，它的约化导致了新的非线性常微分方程组的研究，其解具有特殊的性质。在一般情况下，特例是经典的Darboux-Halphen系统和Chazy方程。这些系统的解决方案与模/自同构函数有关；而在Chazy的情况下，它与众所周知的Ramanujan函数有关。涉及SDYM新削减的研究将继续进行。对微分-差分非线性薛定谔(NLS)方程的研究表明，以前导出的标量差分NLS方程的新的向量扩张有解，并且可通过逆散射变换进行可积。标量和向量差分NLS系统在连续极限上归结为具有重要物理意义的NLS方程。我们将研究这个向量差分NLS方程的新的解和性质。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，而定域解不具有这些性质。这项工作是由PI对计算混沌的早期研究推动的。目前的研究表明，这种现象也存在于非线性光学中，并且具有普遍性。这种无限维且可能是普遍存在的混沌动力学将被详细研究。大振幅波动系统的动力学通常被称为非线性波动。与已有丰富而广泛理论的小振幅现象不同，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，如本方案中所描述的，在许多物理应用中是非常重要的。两个例子是水波和非线性光学，包括光纤通信。极稳定的局域非线性波称为孤子，这是一个与本项目的研究密切相关的课题。近年来，非线性光学的研究主要集中在对孤子等局域大幅度脉冲的研究上。这样的脉冲以各种方式使用，例如光束的整形和控制。在光纤通信中，了解大幅值光脉冲的特性对下一代通信系统具有重要意义。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边缘。预计所有新成果都将在主要期刊上发表。

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.