Wave Turbulence: Open Challenges and New Opportunities

NSF奖摘要-DMS-0072803数学科学：波浪湍流--开放的挑战和新的机遇扎哈罗夫波湍流，一种由源和汇驱动远离平衡的弱耦合频散波列的海洋的湍流，重新研究的时机已经成熟。首先，有许多开放的挑战需要解决：离散性在阻碍光谱能量重新分配方面的影响；各向异性有限通量谱在风力驱动的海洋中的一般性质；在强磁场存在下的磁流体动力学中；在声波中；运动的隐藏常量的作用；有限容量Kolmogorov谱的惊人实现方式；费米-狄拉克(和玻色-爱因斯坦)光谱在半导体激光等情况下的修正，其中有限通量效应是重要的。其次，这个主题很重要，因为它提供了易于处理的模型，可以利用这些模型来洞察间歇性现象。由于应用波动湍流理论的近似在波数上一般不一致，因此在高波数和低波数时都有间歇窗口。在这些尺度上，系统是完全非线性的，并表现出与连贯但几乎奇异的结构相关的间歇性活动爆发。我们的目标是建立一个全面的理论，将波动湍流与随机发生的相干事件结合起来。这个项目研究了在广泛的物理系统中出现的非线性场方程复数解的长期统计行为。这项研究的应用包括：风力驱动的海浪、天气、透明材料中的激光光束、声波、太阳风和恒星大气以及半导体激光器中的电荷运动。所有这些系统都表现出具有共同特征的湍流行为，该项目发展了这种共同结构的理论。目标是了解这一类物理系统的行为，并研究其中存在大波动的非线性系统的一般统计行为

NSF Award Abstract - DMS-0072803Mathematical Sciences: Wave Turbulence -- Open Challenges and New OpportunitiesAbstract0072803 ZakharovWave turbulence, the turbulence of a sea of weakly-coupled dispersive wave trainsdriven far from equilibrium by sources and sinks, is ripe for renewed investigation.First, there are many open challenges which need to be addressed: the effectof discreteness in impeding spectral energy redistribution; the general nature ofanisotropic finite flux spectra in wind-driven oceans, in magnetohydrodynamics inthe presence of strong magnetic field, and in sound waves; the role of hidden constantsof the motion; the surprising manner in which finite capacity Kolmogorov spectraare realized; the modification of Fermi-Dirac (and Bose-Einstein) spectra in contextssuch as semiconductor lasing where finite flux effects are important. Second, thesubject is important because it provides tractable models which can be exploitedto gain insight into the phenomenon of intermittency. Because the approximationswhich underlie the applications of wave turbulence theory are generally not valid uniformly in wave number, there are windows of intermittency at both high and low wave number values. At these scales, the system is fully nonlinear and exhibits intermittent bursts of activity associated with coherent but almost singular structures. Our aim is to build a comprehensive theory which combines and connects wave turbulence with the random occurrence of coherent events.This project investigates the long-time statistical behavior of complex solutions to nonlinear field equations arising in a broad family of physical systems. Applications of the research include modeling of: wind-driven ocean waves, weather, laser beams in transparent materials, sound waves, solar wind and stellar atmospheres, and motion of charges in semiconductor lasers. All these systems exhibit turbulent behavior with common features, and the project develops the theory of this common structure. The goal is to gain understanding of the behavior of this family of physical systems as well as to study the general statistical behavior of nonlinear systems in which large fluctuations