Wave Turbulence: A Wealth of Applications and a Rich Paradigm for Turbulent Systems

摘要：DMS-0404577A Newell和V Zakharov波湍流：湍流系统的丰富应用和丰富范例研究场方程的解的统计行为，通常是守恒的和哈密顿的，描述有源和汇存在的弱非线性和色散波系统。它有许多应用，从海洋波浪到磁流体力学和光学。但它不仅因为应用而引起人们的兴趣。它为研究基本的哈密顿偏微分方程长时间解行为提供了一个非平凡的工具。这个理论非常丰富，除了非常罕见的情况外，它并不适用于所有的尺度。我们最近对如何预测随机发生的、健壮的和完全非线性的相干结构的击穿范围有了一个理解。这导致了反常和间歇性的行为，其特征可以在高阶时刻戏剧性地看到，有时甚至在低阶时刻也能看到。这也会导致与高斯统计的很大偏差。由于在某些情况下可以识别和理解随机出现的结构的原因，波浪湍流为研究湍流和非平衡系统的指导原则提供了一个不平凡的范式。波浪湍流有着丰富的应用，也是研究非平衡系统最有用的范式。因此，它是学生和年轻科学家的优秀培训工具，因为许多解和整体性质是可计算的，从而帮助那些新进入该领域的人对中心问题有一个具体的理解。这两位PI在他们的职业生涯中一直与令人敬畏的年轻研究人员联系在一起，并培养了20多名积极贡献于非线性科学领域的博士学位。此外，我们在这项提案中列入了我们计划采取的几项具体行动，以便促进培训新的研究人员，与世界各地的团队合作，并传播新的成果。

Abstract: DMS-0404577A Newell and V ZakharovWave Turbulence: A Wealth of Applications and a Rich Paradigm for Turbulent SystemsWave turbulence is the study of the statistical behavior of solutions of field equations, usually conservative and Hamiltonian, describing weakly nonlinear and dispersive wave systems in the presence of sources and sinks. It has many applications, from ocean waves to magnetohydrodynamics and optics. But it is not only of interest because of applications. It provides a nontrivial tool for examining the long time solution behavior of the underlying Hamiltonian partial di_erential equations. It is extraordinarily rich in that, save for very rare circumstances, the theory is not valid for all scales. We have recently developed an understanding for how to predict the breakdown ranges in whichrandomly occurring, robust and fully nonlinear coherent structures occur. These lead to anomalous and intermittent behaviors whose signatures can be seen dramatically in high order moments and sometimes even in low order ones. It also leads to large deviations from Gaussian statistics. Because in several cases it is possible to identify and understand the reasons for the emergence of the randomly occurring structures, wave turbulence provides a nontrivial paradigm for investigating the guiding principles governing turbulent and nonequilibrium systems in general.Wave turbulence has a wealth of applications and is also a most useful paradigm for the study of nonequilibrium systems. Therefore, it is an excellent training tool for students and young scientists because many solutions and bulk properties are calculable, thus helping those new to the field to gain a concrete understanding of central issues. The two PI's have been linked with formidable stables of young researchers throughout their professional lives and have produced more than 20 Ph.D's who are actively contributing to the field of nonlinear science. Further, we include in this proposal several concrete actions we plan to take in order to facilitate the training of new researchers, to collaborate with teams throughout the world and to disseminate new results.